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UNITS AND 
PHYSICAL CONSTANTS. 



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UNITS AND 
PHYSICAL CONSTANTS. 



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£7^0 ' • r. 



UNITS AND 



PHYSICAL CONSTANTS. 



BY 

J. D. JgVERETT, M.A., D.O.L., F.R.S., F.R.S.E., 

PROFESSOR OF NATURAL PHILOSOPHY IN QUEEN'S COLLEGE, BELFAST. 



SECOND EDITION. 

\ 



|Cotti0n : 
MAOMILLAN AND CO., 

AND NEW YORK. 
, 1886. ^ 

[The riffht of transMion and r^produdioa -U rewrot-^A 



I 



I 



I! 



I THE NEW YORK 

I PUBLIC LIBRARY 

I 825920 

1^ ASTOR. LENOX AND 
i'LDEN FOUNDATION.- 

1918 L 






-iiTTrri 



OLASGOW : PRINTED AT THB UNIVERSITY PBE88 
BT ROBERT MACLEHOSE. 



PREFACE TO "ILLUSTRATIONS OF THE O.G.S. 
SYSTEM OF UNITS," PUBLISHED BY THE 
PHYSICAL SOCIETY OF LONDON IN 1875. 

The quantitative study of physics, and especially of the 
relations between different branches of physics, is every 
day receiving more attention. 

To facilitate this study, by exemplifying the use of a 
system of units fitted for placing such relations in the 
clearest light, is the main object of the present treatise. 

A complete account is given of the theory of units ah 
initio. The Oentimetre-Gramme-Second (or O.G.S.) sys- 
tem is then explained ; and the remainder of the work is 
occupied with illustrations of its application to various 
branches of physics. As a means to this end, the most 
important experimental data relating to each subject are 
concisely presented on one uniform scale — a luxury 
hitherto unknown to the scientific calculator. 

I am indebted to several friends for assistance in special 
departments — ^but especially to Professor Clerk Maxwell 
and Professor G. 0. Foster, who revised the entire manu- 
script of the work in its original form. 

Great pains have been taken to make the work correct 
as a book of reference. Readers who may discover any 
errors will greatly oblige me by pointing them out. 



PREFACE TO FIJEIST EDITION OF UNITS 
AND PHYSICAL CONSTANTS, 1879. 

This Book is substantially a new edition of my *• Illus- 
trations of the C.G.S. System of Units " published in 1875 
by the Physical Society of London, supplemented by an 
extensive collection of physical data. The title has been 
changed with the view of rendering it more generally 
intelligible. 

Additional explanations have been given upon some 
points of theory, especially in connection with Stress and 
Strain, and with Coefl&cients of Diffusion. Under the 
former head, I have ventured to introduce the terms 
"resilience" and "coefficient of resilience," in order to 
avoid the multiplicity of meanings which have become 
attached to the word "elasticity." 

A still greater innovation has been introduced in an 
extended use of the symbols and processes of multiplication 
and division, in connection with equations which express 
not numerical but physical equality. The advantages of 
this mode of procedure are illustrated by its application 
to the solution of the most difficult problems on units that 
I have been able to collect from standard text-books 
(chiefly from Wormell's * Dynamics '). 

I am indebted to several friends for contributions of 
experimental data. 



viii PREFACE TO FIRST EDITION. 

A Dutch translation of the first edition of this work 
has been made by Dr. C. J. Matthes, Secretary of the 
Boyal Academy of Sciences of Amsterdam, and was pub- 
lished in that city in 1877. 

Though the publication is no longer officially connected 
with the Physical Society, the present enlarged edition 
is issued with the Society's full consent and approval. 



PREFACE TO THE PRESENT EDITION. 

In collecting materials for this edition, I have gone care- 
fully through the Transactions and Proceedings of the 
Royal Society, the Royal Society of Edinburgh, and the 
Physical Society of London, from 1879 onwards, besides, 
consulting numerous papers, both English and foreign, 
which have been sent to me by their authors. I have 
also had the advantage of the co-operation of Dr. Pierre 
Chappuis (of the Bureau International des Poids et 
Mesures), who has for some years been engaged in pre- 
paring a German edition. Several items have been ex- 
tracted from the very elaborate and valuable Physikalisch- 
Gheinische Tabellen of Landolt and Bbrnstein (Julius 
Springer, Berlin, 1883). Among friends to whom I am 
indebted for data or useful suggestions, are Prof. Barrett, 
Mr. J. T. Bottomley, Prof. G. C. Foster, Prof. Lodge, 
Prof. Newcomb, Mr. Preece, and the Astronomer-Royal. 

The expository portions of the book are for the most 
part unchanged; but a Supplemental Section has been 
added (p. 34) on physical deductions from the dimensions 
of units ; a simplification has been introduced in the dis- 
cussion of adiabatic compression (p. 1 25) ; and the account 
of thermoelectricity (p. 172) has been re-written and 
enlarged. The name "thermoelectric height" has been 
introduced to denote the element usually represented by 
the ordinates of a thermoelectric diagram. 



X PREFACE TO THE PRESENT EDITION. 

The preliminary " Tables for reducing other measures 
to C.G.S. measures" have been greatly extended, and in 
each case the reciprocal factors are given which serve for 
reducing from C.G.S. measures to other measures. Pro- 
fessor Miller's comparison of the kilogramme and pound 
is supplemented by three later comparisons officially made 
at the Bureau International, 

A nearly complete list of the changes and additions 
now introduced is appended to this Preface, as it will 
probably be useful to possessors of the previous edition. 

The adoption of the Centimetre, Gramme, and Second, 
as the fundamental units, by the International Congress 
of Electricians at Paris in 1881, led to the immediate 
execution of a French translation of this work, which was 
published at Paris by Gauthier-Yillars in 1883. The 
German translation was commenced about the same time, 
but the desire to perfect its collection of physical data has 
caused much delay. It will be brought out by Ambrosius 
Barth, the publisher of Wiedemann's Annalen. A Polish 
edition, by Prof. J. J. Boguski, was published at Warsaw 
in 1885 ; and permission has been asked and granted for 
the publication of an Italian edition. 

J. D. EVERETT. 

Belfast, September, 1886. 



LIST OF CHANGES AND ADDITIONS. 

PAOEB 

Tables for conversion to and from C.G.S., - - - 1-4 

Formula for gr, 26 

" Watt " defined, 30 

Physical deductions from dimensions, .... 34-37 

Specific gravity table, 40 

Surface tension of liquids, 49, 50 

Thickness of soap films, 50, 51 

Poisson's ratio, 62 

Velocity of Hght, 76 

Indices of refraction of crystals, etc. , - - - - 80, 81 

Refraction and dispersion of gases, 83-85 

Rotation by quartz, 85 

Candle, carcel, etc., 86 

Specific heat, 90-94 

Melting, 95-97 

Boiling, 98 

Pressure of steam from 0° to 150*, 102 

Critical points of gases, 103 

Conductivity (thermal) of solids, - • - - 115, 116 

„ ,, of liquids, 117 

Joule's equivalent, 121 

Adiabatic compression, 125, 126 

Expansion of mercury, 129 

Collected data for air, 129 

Density of moist air, 130 

Magnetic susceptibility, 133 

Greenwich magnetic elements, 138 

Magneto-optic rotation, 139 

Ratio of the two units of electricity, .... 145 

Specific inductive capacity, 148-I5ft 



xii CHANGES AND ADDITIONS. 

PA 

Practical units, 151, 

Resolutions of Congress and Conference, - • - 153, 

Resistance, 15^ 

Gauge and resistance of wires, 165. 

Electro-motive force of cells, 167, 

Thermoelectricity, 175 

Electrochemic€il equivalents, - • - - - - 17S 

Heat of combination of cells, - ! - - - - 181, 

Compression of liquids, 

Expression of decimal multiples, etc., .... 



CONTENTS. 

PAGES 

Tables for Reducing to and from C.G.S. Measures, 1-4 

Chapter L — General Theory of Units, - - - 5-18 

Chapter II. — Choice of Three Fundamental Units, - 19-24 

Chapter HI. — ^Mechanical Units, 25-34 

Supplemental Section, on Physical Deductions from 

Dimensions, 34-37 

Chapter IV.— Hydrostatics, 38-51 

Chapter V. — Stress, Strain, and Resilience, • - 52-64 

Chapter VI. — Astronomy, 65-69 

Chapter VII.— Velocity of Sound, .... 70-74 

Chapter Vin.— Light, 75-86 

Chapter IX.— Heat, 87-130 

Chapter X.— Magnetism, 131-139 

Chapter XL —Electricity, 140-188 

Omission and Suggestion, 189, 190 

Appendix.— Reports of Units Cemmittee of British 

Association, - - 191-195 

Index, 196-200 



UNITS AND PHYSICAL CONSTANTS. 



TABLES FOR REDUCING TO AND FROM 

C.a.S. MEASURES. 



The abbreviation cm, is used for centimetre or centimetres, 

gm,, „ gramme or grammes^ 

c.c, „ cubic centimetre(8). 

The numbers headed " reciprocals " are the factors for 
reducing ^om C.G.S. measures. 

Length, 





cm. 


Reciprocals. 


1 inch, - 


2-5400 


•39370 


1 foot, - 


30-4797 


•032809 


1 yard,- 


91-4392 


•010936 


1 mile, 


= 160933 


6-2138 xlO-« 


1 nautical mile, - 


= 185230 


5-398 xlO-« 



More exactly, according to Captain Clarke's compari- 
sons of standards of length (printed in 1866), the metre is 
equal to 1-09362311 yard, or 3-2808693 feet, or 39-370432 
inches, the standard metre being taken as correct at 0° C, 
and the standard yard as correct at 16f'' C. Hence the 
inch is 2^5399772 centimetres. 



Area, 



1 sq. inch, 
1 sq. foot, 
1 sq. yard, 
1 sq. mile, 

i) 



sq. cm. 


Reciprocals. 


6-4516 


•1550 


929 01 


•001076 


8361-13 


•0001196 


2-59xl0i« 


3-861 X 10" 


A 





2 



UNITS AND PHYSICAL CONSTANTS. 



Volume. 





cub. cm. 


Reciprocals. 


I cubic inch, 


- -= 16-387 


•06102 


1 cubic foot, 


- - 28316- 


3-532x10-6 


1 cubic yard, 


- -764535- 


1-308 xl0-« 


1 pint, - 


- = 567-63 


-001762 


] gallon, 


- = 4541- 
Mass, 


•0002202 




gm. 


Reciprocals. 


1 grain, 


- - -0647990 


15-432 


I ounce avoir.. 


- - 28-3495 


•035274 


1 pound ,, 


- =453-59 


-0022046 


1 ton, - 


■ = 1-01605 xlO« 


9-84206x10-7 



According to the compaidson made by Professor W. H. 
Miller in 1844 of the "kilogramme des Archives," the 
standard of French weights, with two English pounds of 
platinum, and additional weights, also of platinum, the kilo- 
gramme is 15432-34874 grains, of which the new standard 
})Ound contains 7000. Hence the kilogramme would be 
2-2046212 pounds, and the pound 453^59265 grammes. 

Three standard pounds, one of platinum-iridium and 
the other two of gilded bronze, belonging to the Standards 
Department^ were compared, in 1883, at the Bureau In- 
ternational des Poids et MesureSy with standards belong- 
ing to the Bureau, and their values in grammes were 
found to be respectively 

453-59135, 

453-58924, 

45358738. 
— Travaux et Memoires, tome IV. 



Velocity, 



1 foot per second, 

1 statute mile per hour, 

1 nautical mile per hour, 

1 kilometre per hoar^ 



cm. per sec. 
= 30-4797 
=44 704 
=51 453 

=27-777 



Reciprocals. 
•032809 
•022369 
•019435 

•036 



TABLES. 3 



Acceleration, 



cm. per sec. per sec. Reciprocal. 
1 ft. per sec. per sec, - =30-4797 032809 

JDensiti/, 

gm. per c.c. Reciprocals. 

^T»JI^^!^'ilHr**"™ "M =1000013 -999987 

maximum density, - - ) 

I lb. per cubic foot, - - = 016019 62*426 

1 grain per cubic inch, - - := 003954 252-88 

Stress (in gravitation measure), 

gm. per sq. cm. Reciprocals. 

1 lb. per sq. foot, - = '48826 2 0481 
1 lb. per sq. inch, - =:: 70-31 -014223 

1 inch of mercury at| ^ 3^.53^ ..^^^^^ 

30 inches „ ,, =1036-0 00096525 

760 mm. „ „ =1033-3 -00096777 

Surface Tension (in gravitation measure), 

gm. per cm. Reciprocals. 

1 ipi-ain per linear inch, - = -02551 39-20 

lib. „ foot, - =14-88 -06720 

Work (in gravitation measure), 

gm.-cm. Reciprocals. 

. 1 foot-pound, - =13825 7-2331 xlO-^ 
1 f oot-grain, - = 1 -976 50632 

1 foot-ton, - - = 3097x107 6-494x10-* 
I kilogrammetre, - = 10** 10" ** 

Rate of Working (in gravitation msasure), 

gm.-cm. per sec. Reciprocals. 

1 horse-power, - = 7 -604 x 10» 1 3151 x 10 - ^ 

1 force-de-cheval, - = 7*5x106 1-3333x10-7 

Heat (in gravitation measure), 

gm.-cm. Reciprocals. 

Igm. deg., - - =42400 2-36xl0-» 

1 lb. deg. Cent., - = 1 -923 x 10^ 5-2 x lO-s 

1 „ Fahr., - = 1068x107 9*36x10-8 

The following reductions of gravitation measures to 
absolute measures are on the assumption that ^ = 981 : — 



UNITS AND PHYSICAL CONSTANTS. 



Weight of 1 gm. , - - = 

1 kilogm., - = 

1 tonDe, - = 

1 ton, - — 

1 cwt. , - • = 

1 lb. avoir., - = 
1 oz. 
1 grain, 



Force (in absolute measure). 

Dynes. 



»» 



981 
9*81 X 10« 
9-81 X 10» 
9-97 X 10« 
4-98x107 
4-45 X 105 
2-78 X 10* 
63-57 



1 poundal, 



= 13825 



Reciprocals. 
•001019 
1019xl0-« 
l-019xl0-» 
1-003 xlO-» 
2-008 X 10 -» 
2-247 X 10 -« 
3-697 X 10-« 

-01573 
7-2333x10-* 



(The ratio of the poundal to the dyne is independent of g, ) 
Stress (in absolute measure). 

Dynes per sq. cm. 
1 lb. per sq. foot, • = -479 

lib. „ inch, - = 6-9x10* 
1 gm. „ cm., - = -981 

Ikilo. „ decim., - = 9*81 xlO» 
1 cnv of mercury at 0°C., =13338' 
76 „ „ „ = 1-0136x108 

linch „ „ = 3-388x10* 

30 „ ,. „ = 1-0163 xlO« 



>» 



)) 



»» 



Beciprocals. 

•00209 
1-45x10-5 

•00102 
102x10-* 

•0000736 
9-866 X 10-?^ 
2-95x10-* 
9-84x10-7 

' Surface Tension (in absolute msasure). 

Dynes per cm. Reciprocals. 
1 gm. per linear cm., - - = 981 -00102 

1 grain ,, inch, - - = 25 '04 

lib. „ foot, - - =14600 6-85x10-* 

Work and Energy (in absolute measure). 

Ergs. Reciprocals. 

Icm. cm.,- - = 981 -001019 

1 kilogrammetre, = 9 -81 x 10^ 1 -019 x 10-8 

1 foot-pound, - = 1-356x107 7-37x10-8 

1 foot-poundal, - =421390 2-3731 x 10-« 
(The ratio of the ft. -poundal to the erg is independent of g,) 
I joule - - = 107 ergs. 

Mate of Working (in absolute measure). 



Ergs per sec. 
=7-46x109 
= 7-36x109 
= lO'^ 



1 horse-power, 

1 force-de-cheval, - 

1 watt, - - - . 

Heat (in absolute measure). 

Ergs. 
Igm. deg., - - - =4-2x10'^ 
1 lb. deg. Cent. , - - = 1 -905 x lO^o 
1 ,. Fahr., - - =1 058x1010 



»» 



Reciprocals. 
1-34x10-19 
1-36x10-19 
10-7 



Reciprocals. 
2-38x10-8 
5-25x10-11 
9-45x10-11 



CHAPTER I. 

GENERAL THEORY OF UNITS. 
Units and Derived Units, 

1. The numerical value of a concrete quantity is its 
ratio to a selected magnitude of the same kind, called 
tTie unit. 

Thus, if L denote a definite length, and I the unit 

length, •=- is a ratio in the strict Euclidian sense, and is 

V 

called the numerical value of L. 

The numerical value of a concrete quantity varies 
directly as the concrete quantity itself, and inversely as 
the unit in terms of which it is expressed. 

2. A unit of one kind of quantity is sometimes defined 
by reference to a unit of another kind of quantity. For 
example, the unit of area is commonly defined to be the 
area of the square described upon the unit of length ; 
and the unit of volume is commonly defined as the volume 
of the cube constructed on the unit of length. The units 
of area and volume thus defined are called derived units, 
and are more convenient for calculation than indepen- 
dent units would be. For example, when the above 



6 UNITS AND PHYSICAL CONSTANTS. [chap. 

definition of the unit of area is employed, we can assert 
that [the numerical value of] the area of any rectangle is 
equal to the product of [the numerical values of] its 
length and breadth ; whereas, if any other unit of area 
were employed, we should have to introduce a third factor 
which would be constant for all rectangles. 

3. Still more frequently, a unit of one kind of quantity 
is defined by reference to two or more units of other 
kinds. For example, the unit of velocity is commonly 
defined to be that velocity with which the unit length 
would be described in the unit time. When we specify 
a velocity as so many miles per hour, or so many^^e^ per 
second, we in effect employ as the unit of velocity a mile 
per hour in the former case, and a foot per second in the 
latter. These are derived units of velocity. 

Again, the unit acceleration is commonly defined to 
be that acceleration with which a unit of velocity would 
be gained in a unit of time. The unit of acceleration is 
thus derived directly from the units of velocity and time, 
and therefore indirectly from the units of length and 
time. 

4. In these and all other cases, the practical advantage 
of employing derived units is, that we thus avoid the intro- 
duction of additional factors, which would involve needless 
labour in calculating and difficulty in remembeiing.* 

5. The correlative term to derived is fundamental. 
Thus, when the units of area, volume, velocity, and 

* An example of such needless factors may be found in the rules 
commonly given in English books for findmg the mass of a body 
when its volume and material are given. ** Multiply the volume 
in cubic feet by the specific gravity and by 62*4, and the product 
will be the mass in pounds ; " or ** multiply the volume in cubic 



I.] GENERAL THEORY OF UNITS. 7 

acceleration are defined as above, the units of length and 
time are called the fundamental units. 

Dimensions, 

6. Let us now examine the laws according to which 
derived units vary when the fundamental units are 
changed. 

Let V denote a concrete velocity such that a concrete 
length L is described in a concrete time T ; and let v, Z, t 
denote respectively the unit velocity, the unit length, and 
the unit time. 

The numerical value of V is to be equal to the numerical 
value of L divided by the numerical value of T. But 

these numerical values are — , — , -: 

V I t 

hence we must have 

l^h' (1) 

This equation shows that, when the units are changed 
(a change which does not affect V, L, and T), v must 
vary directly as I and inversely &s t; that is to say, the 
unit of velocity varies directly as the unit of lengthy and 

inversely as the unit of time. 

Y 

Equation (1) also shows that the numerical value — of 

V 

a given velocity varies inversely as the unit of lengthy and 
directly as the unit of time, 

inches by the specific gravity and by 253, and the product will 
be the mass in grains." The factors 62*4 and 253 here employed 
would be avoided — that is, would be replaced by unity, if the 
unit volume of water were made the unit of mass. 



8 UNITS AND PHYSICAL CONSTANTS. [chap. 

7. Again, let A denote a concrete acceleration such 
that the velocity V is gained in the time T', and let a 
denote the unit of acceleration. Then, since the numerical 
value of the acceleration A is the numerical value of the 
velocity V divided by the numerical value of the time T, 
we have 

It t V 
But by equation (1) we may write —for—. We 



I T V 



thus obtain 

A^L t^ t 
a~JfT' 



(2) 



This equation shows that when the units a, l^ t are 
changed (a change which will not affect A, L, T or T'), a 
must vary directly as l^ and inversely in the duplicate 

ratio of t ; and the numerical value — will vary inversely 

a 

as I, and directly in the duplicate ratio of t In other 

words, the unit of acceleration varies directly as the unit 

of lengthy and inversely as tJie square of the unit of time; 

and the numerical value of a given acceleration varies 

inversely as the vm,it of lengthy and directly as the square 

of the unit of time. 

It will be observed that these have been deduced as 
direct consequences from the fact that [the numerical 
value of] an acceleration is equal to [the numerical 
value of] a length, divided by [the numerical value 
of] a time, and then again by [the numerical value of] 
a time. 

The relations here pointed out are usually expressed by 



1.] GENERAL THEORY OF UNITS. 9 

lArtrrfVi 

saving that the dimensions of acceleration* are .-r-^ — , or 

(time)- 

that the dimensions o/t/ie unit of acceleraiion* are 

unit of length 
(unit of time)2' 

8. We have treated these two cases very fully, by way 
of laying a firm foundation for much that is to follow. 
We shall hereafter use an abridged form of reasoning, 
«uch as the following : — 

, .. length 
velocity = — r^- : 

•^ time ' 

, .. velocity length 

acceleration = — ; ^ = , . .^ » 

time (time)2 

Such equations as these may be called dimensional 
eqvuitions. Their full interpretation is obvious from what 
precedes. In all such equations, constant numerical factors 
can be discarded, as not affecting dimensions. 

9. As an example of the application of equation (2) we 
shall compare the unit acceleration based on the foot and 
second with the unit acceleration based on the yard and 
minute. 

Let I denote a foot, L a yard, t a second, T a minute, 
Ty a minute. Then a will denote the unit acceleration 
based on the foot and second, and A will denote the unit 

♦Professor James Thomson (*Brit. Assoc. Report,* 1878, p. 
452) objects to these expressions, and proposes to substitute the 
following : — 

"Change-ratio of unit of acceleration =;^^^j^^/' 

This is very clear and satisfactory as a full statement of the 
meaning intended ; but it is necessary to tolerate some abridg- 
ment of it for practical working. 



10 UNITS AND PHYSICAL CONSTANTS. [chap, 

acceleration based on the yaid and minute. Equation 
(2) becomes 

a 1 "" V60/ 1200 ' ^ ^ 

that is to say, an acceleration in which a yard per minute 

of velocity is gained per minute, is A^ of an acceleration 

in which a foot per second is gained per second. 

Meaning of ^^per" 

10. The word per, which we have several times em- 
ployed in the present chapter, denotes division of the 
quantity named before it by the quantity named after it. 
Thus, to compute velocity in feet per second, we must 
divide a number of feet by a number of seconds.* 

If velocity is continuously varying, let x be the number 
of feet described since a given epoch, and t the number 

doc 
of seconds elapsed, then - is what is meant by the 

at 

number of feet per second. The word should never be 
employed in the specification of quantities, except when 
the quantity named before it varies directly as the quantity 
named after it, at least for small variations — as, in the 
above instance, the distance described is ultimately pro- 
portional to the time of describing it. 

Extended Sense of the terms ** Multiplication " and 

" Division" 

11. In ordinary multiplication the multiplier is always- 

* It is not correct to speak of interest at the rate of Fim Pounds 
per cent. It should be simply Five per cent. A rate of five pounds 
in every hundred pounds is not different from a rate of five- 
shillings in every hundred shillings. 



I.] GENERAL THEORY OF UNITS. H 

a mere numerical quantity, and the product is of the same 
nature as the multiplicand. Hence in ordinary division 
either the divisor is a mere numerical quantity and the 
quotient a quantity of the same nature as the dividend ; 
or else the divisor is of the same nature as the dividend^ 
and the quotient a mere numerical quantity. 

But in discussing problems relating to units, it is con- 
venient to extend the meanings of the terms " multiplica- 
tion " and " division." A distance divided by a time 
will denote a velocity — the velocity with which the given 
distance would be described in the given time. The dis- 
tance can be expressed as a unit distance multiplied by a 
numerical quantity, and varies jointly as these two factors ; 
the time can be expressed as a unit time multiplied by a 
numerical quantity, and is jointly proportional to these two 
factors. Also, the velocity remains unchanged when the 
time and distance are both changed in the same ratio. 

1 2. The three quotients 

1 mile ' 5280 ft. 22 ft. 
I hour 3600 sec' 15 sec. 

all denote the same velocity, and are therefore to be 
regarded as equal. In passing from the first to the 
second, we have changed the units in the inverse ratio 
to their numerical multipliers, and have thus left both 
the distance and the time unchanged. In passing from 
the second to the third, we have divided the two numeri- 
cal factors by a common measure, and have thus changed 
the distance and the time in the same ratio. A change 
in either factor of the numerator will be compensated 
by a proportional change in either factor of the denom- 
inator. 



1 2 UNITS AND PHYSICAL CONSTANTS. [chap. 

09 £«. ^ 22 
Further, since the velocity - ~ — - is — of the velo- 

ID sec. 10 

.. 1 ft. ^.^, , ^ ..22 ft. 22 ft. 

<;ity , we are entitled to write = 7^ . , 

1 sec. 15 sec. 15 sea 

thus separating the numerical part of the expression from 

the units part. 

In like manner we may express the result of Art. 9 by 

writing 

yard _ 1 foot 

(minute)2 " 1200 * (second)^' 

Such equations as these may be called *' physical 
-equations," inasmuch as they express the equality of 
physical quantities, whereas ordinary equations express 
the equality of mere numerical values. The use of 
physical equations in problems relating to units is to be 
strongly recommended, as affording a natural and easy 
clue to the necessary calculations, and especially as 
obviating the doubt by which the student is often 
-embarrassed as to whether he ought to multiply or 
divide. 

13. In the following examples, which illustrate the use 
of physical equations, we shall employ I to denote the 
unit length, m the unit mass, and t the unit time. 

Ex. 1. If the yard be the unit of length, and the 
acceleration of gravity (in which a velocity of 32*2 ft. per 
«ec. is gained per sec.) be represented by 2415, find the 
unit of time. 

We have I = yard, and 

32-2 J^, = 2415 { = 2415 ^' 
(sec.)2 fi t^ 

.-. t^ = ^^1^ sec.* = 225 secS t = 15 sec. 



I.] GENERAL THEORY OP UNITS. 15 

Ex. 2. If the unit time be the second, the unit density 
162 lbs. per cub. ft., and the unit force* the weight of an 
ounce at a place where g (in foot-second units) is 32^ 
what is the unit length ? 

We have < = sec, J. = 162^3, 

and -^ = 32 . ?5l^, or ml^^ 32 oz. ft. = 2 lb. ft. 
sec* sec. 

Hence by division 

^* = ^ (ft.)*. ? = J ft. = 4 in. 

Ex. 3. If the area of a field of 10 acres be represented 
by 100, and the acceleration of gravity (taken as 32 foot- 
second units) be 58 1, find the unit of time. 

We have 48400 (yd.)^=100 V, whence Z = 22 yd.; 

and 32-A. = 58| -4 = ^ ^> 

(sec.)^ ^ f 3 t' ' 

whence f = — — sec.^ =121 sec.^ ^=11 sec, 

Ex. 4. If 8 ft. per sec, be the unit velocity, and the 
acceleration of gravity (32 foot-second units) the unit 
acceleration, find the units of length and time. 

We have the two equations 

I Q it. L QO it. 

t sec. r sec* 

whence by division t = \ sec, and substituting this value 
of ^ in the first equation, we have 4 ?= 8 ft., Z= 2 ft. 

Ex. 6. If the unit force be 100 lbs, weight, the unit 
length 2 ft., and the unit time \ sec, find the unit mass, 
the acceleration of gravity being taken as 32 foot-second 
units. 

* For the dunensions of density and force, see Art 14. 



14 UNITS AND PHYSICAL CONSTANTS. [chap. 

We have 1 = 2 ft., < = J see., 

ft. ml m 2 it. 



TOO lb. 32 



sec." V TW" sec. 



tliat is 100 X 32 lb. =^2m,m = 100 lb. 

Ex. 6. The number of seconds in the unit of time is 
vqual to the number of feet in the unit of length, the unit 
of force is 750 lbs. weight [g being 32], and a cubic foot 
of the standard substance [substance of unit density] con- 
tains 13500 oz. Find the unit of time. 

Let t = x sec, then l = xft; also let m — y lb. Then 
we have 

ml _ y lb. xii. _ y lb. ft . __ --^ 09 lb. ft. 
t^ ^ sec.* X sec.*^ sec* 



or 



V- = 750 X 32. 



X 



whence \ = 13500 x }-, 

a^ 16 

Hence by division 

2 _ 750 X 32 X 16 __ 16* _ 16 , _ 16 
^ 13500 3^' "^ - T* ' " T'^"^ 

Ex. 7. When an inch is the unit of length and t the 
unit of time, the measure of a certain acceleration is ay 
when 5 ft. and 1 min. are the units of length and time 
respectively, the measure of the same acceleration is 10 a. 
Find t 

Equating the two expressions for the acceleration, we 

, inch TA 5 ft. 

have a --^ = 10 a . , 

t^ (mm.)* 



i.] GENERAL THEORY OF UNITS. 15 

I j^ t ' \'> inch (min.)*'' ^ / v., 

whence t' = (min.)- ,^ . = J- = 6 (sec.)'' 

^ ' 50 ft. 600 ^ ^ 

< = x/6 sec. 

Ex. 8. The numerical value of a certain force is 56 

when the pound is the unit of mass, the foot the unit of 

length, and the second the unit of time ; what will be the 

numerical value of the same force when the hundredweight 

is the unit of mass, the yard the unit of length, and the 

minute the unit of time ? 

Denoting the required value by x we have 

p.« lb. ft. _ cwt. yard 
00 — — X : — - — ; 

sec.- mm.'* 



a; = 56 ^^- ^*- 



(min.y 
sec./ 



cwt. yd. 

= 56 X -1- X 1 X 60^ = 600. ^ 
112 3 

Dimensions of Mechanical and Geometrical Quantities. 

14. In the following list of dimensions, we employ the 
letters L, M, T as abbreviations for the words Length, 
MasSj Time, The symbol of equality is used to denote 
sameness of dimensions. 

Area = L^, Volume = L^, Velocity = -, 

Acceleration = =5, Momentum = -— . 

M 

Density = =-, density being defined as mass per unit 

1-r 

volume. 

Force = -=^, since a force is measured by the momen- 
tum which it generates per unit of time, and is therefoi'e 



16 UNITS AND PHYSICAL CONSTANTS. [chap, 

the quotient of momentum by time — or since a force i& 
measured by the product of a mass by the acceleration 
generated in this mass. 

Work = -7=5-, being the product of force and distance. 

ML* 
Kinetic energy - -7p^, being half the product of mass 

by square of velocity. The constant factor J can be 

omitted, as not affecting dimensions. 

ML* 
Moment of couple = -^pj , being the product of a force 

by a length. 

The dimensions of angle,* when measured by — - — , 

radius 

are zero. The same angle will be denoted by the same 

number, whatever be the unit of length employed. In 

fact we have — r^— = ~ = Ifi, 
radius L 

The work done by a couple in turning a body through 

any angle, is the product of the couple by the angle. 

The identity of dimensions between work and couple is 

thus verified. 

Angular velocity - -. 

Angular acceleration = — . 

Moment of inertia = ML*. 

ML* 

Angular momentum = moment of momentum = , 

* The name radian has been given to the angle whose arc is 
equal to radius. "An angle whose value in circular measure is 
0** iB^an angle of radians.** 



1.] GENERAL THEORY OF UNITS. 17 

being the product of moment of inertia by angular velo- 
city, or the product of momentum by length. 

Intensity of pressurej or intensity of stress generally, 

being force per unit of area, is of dimensions ; that 

area 

. M 

IS 



Intensity of force of attraction at a point, often called 
simply force at a point, being force per unit of attracted 

mass, is of dimensions or =5. It is numerically 

mass T^ 

equal to the acceleration which it generates, and has 

accordingly the dimensions of acceleration. 

The absolute force of a centre of attraction, better called 

the strength of a centre, may be defined as the intensity of 

force at unit distance. If the law of attraction be that 

of inverse squares, the strength will be the product of the 

intensity of force at any distance by the square of this 

Curvature (of a curve) = — , being the angle turned by 

the tangent per unit distance travelled along the curve. 
Tortuosity = — , being the angle turned by the osculat- 

ing plane per unit distance travelled along the curve. 

The solid angle or aperture of a conical surface of any 
form is measured by the area cut off by the cone from a 
sphere whose centre is at the vertex of the cone, divided 
by the square of the radius of the sphere. Its dimensions 
are therefore zero ; or a solid angle is a numerical quan- 
tity independent of the fundamental units. 

B 



18 UNITS AND PHYSICAL CONSTANTS, [chap. i. 

The specific cv/rvatvAre of a surface at a given point 
(Grauss's measure of curvature) is the solid angle de- 
scribed by a line drawn from a fixed point parallel to the 
normal at a point which travels on the surface round the 
given point, and close to it, divided by the very small 

area thus enclosed. Its dimensions are therefore — .. 

The mean curvature of a surface at a given point, in 
the theory of Capillarity, is the arithmetical mean of the 
curvatures of any two normal sections normal to each 

other. Its dimensions are therefore -=-. 

Li 



19 



CHAPTER II. 

CHOICE OF THREE FUNDAMENTAL UNITS. 

15. Nearly all the quantities with which physical 
science deals can be expressed in terms of three funda- 
mental units ; and the quantities commonly selected to 
serve as the fundamental units are 

a definite length, 

a definite mass, 

a definite interval of time. 

This particular selection is a matter of convenience 
rather than of necessity ; for any three independent units 
are theoretically sufficient. For example, we might em- 
ploy as the fundamental units 

a definite mass, 

a definite amount of energy, 

a definite density. 

16. The following are the most important considera- 
tions which ought to guide the selection of fundamental 
units : — 

(1) They should be quantities admitting of very 
accurate comparison with other quantities of the same 
kind. 



20 UNITS AND PHYSICAL CONSTANTS. [chap. 

(2) Such comparison should be possible at all times. 
Hence the standards must be permanent — that is, not 
liable to alter their magnitude with lapse of time. 

(3) Such comparisons should be possible at all places. 
Hence the standards must not be of such a nature as 
to change their magnitude when carried from place to 
place. 

(4) The comparison should be easy and direct. 
Besides these experimental requirements, it is also 

desirable that the fundamental units be so chosen that 
the definition of the various derived units shall be easy, 
and their dimensions simple. 

17. There is probably no kind of magnitude which so 
completely fulfils the four conditions above stated as a 
standard of masSy consisting of a piece of gold, platinum, 
or some other substance not liable to be affected by 
atmospheric influences. The comparison of such a 
standard with other bodies of a])proximately equal 
mass is effected by weighing, which is, of all the 
operations of the laboratory, the most exact. Very ac- 
curate copies of the standard can thus be secured; and 
these can be carried from place to place with little risk 
of injury. 

The third of the requirements above specified forbids 
the selection of a force as one of the fundamental 
units. Forces at the same place can be very accurately 
measured by comparison with weights; but as gravity 
varies from place to place, the force of gravity upon a 
piece of metal, or other standard weight, changes its 
magnitude in travelling from one place to another. A 
spring-balance, it is true, gives a direct measure of 



n.] THKEE FUNDAMENTAL UNITS. 21 

force; but its indications are too rough for purposes of 
accuracy. 

18. Length is an element which can be very accurately 
measured and copied. But every measuring instrument 
is liable to change its length with temperature. It is 
therefore necessary, in defining a length by reference to a 
concrete material standard, such as a bar of metal, to 
state the temperature at which the standard is correct. 
The temperature now usually selected for this purpose is 
that of a mixture of ice and water (0° C), observation 
having shown that the temperature of such a mixtiire is 
constant. 

The length of the standard should not exceed the length 
of a convenient measuring-instrument ; for, in comparing 
the standard with a copy, the shifting of the measuring- 
instrument used in the comparison introduces additional 
risk of error. 

In endrstanda/rdSy the standard length is that of the bar 
as a whole, and the ends are touched by the instrument 
every time that a comparison is made. This process is 
liable to wear away the ends and make the standard false. 
In Ime-standcurds, the standard length is the distance be- 
tween two scratches, and the comparison is made by 
optical means. The scratches are usually at the bottom 
of holes sunk halfway through the bar. 

19. Time is also an element which can be measured 
with extreme precision. The direct instruments of mea- 
surement are clocks and chronometers; but these are 
checked by astronomical observations, and especially by 
transits of stars. The time between two successive tran- 
sits of a star is (very approximately) the time of the 



22 UNITS AND PHYSICAL CONSTANTS. [chap. 

earth's rotation on its axis ; and it is upon the uniformity 
of this rotation that the preservation of our standards of 
time depends. 

Necessity for a Common Scale, 

20. The existence of quantitative correlations between 
the various forms of energy, imposes upon men of science 
the duty of bringing all kinds of physical quantity to one 
common scale of comparison. Several such measures 
(called absolute measures) have been published in recent 
years; and a comparison of them brings very promi- 
nently into notice the great diversity at present existing 
in the selection of particular units of length, mass, and 
time. 

Sometimes the units employed have been the foot, the 
grain, and the second ; sometimes the millimetre, milli- 
gramme, and second ; sometimes the centimetre, gramme, 
and second; sometimes the centimetre, gramme, and 
minute ; sometimes the metre, tonne, and second ; some- 
times the metre, gramme, and second ; while sometimes a 
mixture of units has been employed ; the area of a plate, 
for example, being expressed in square metres, and its 
thickness in millimetres. 

A diversity of scales may be tolerable, though undesir- 
able, in the specification of such simple matters as length, 
area, volume, and mass when occurring singly ; for the 
reduction of these from one scale to another is generally 
understood. But when the quantities specified involve 
a reference to more than one of the fundamental units^ 
and especially when their dimensions in terms of these 
units are not obvious, but require 'careful working out. 



n.] THREE FUNDAMENTAL UNITS, 23 

there is great increase of difficultj and of liability to 
mistake. 

A general agreement as to the particular units of length, 
mass, and time which shall be employed — if not in all 
scientific work, at least in all work involving complicated 
references to units — is urgently needed ; and almost any 
one of the selections above instanced would be better than 
the present option. 

21. We shall adopt the recommendation of the Units 
Committee of the British Association (see Appendix), 
that all specifications be referred to the Centimetre^ the 
Gramme, and the Second. The system of units derived 
from these as the fundamental units is called the C.G,S. 
system; and the units of the system are called the CG.S, 
units. 

The reason for selecting the centimetre and gramme, 
rather than the metre and gramme, is that, since a 
gramme of water has a volume of approximately 1 cubic 
centimetre, the former selection makes the density of 
water unity; whereas the latter selection would make 
it a million, and the density of a substance would 
be a million times its specific gravity, instead of being 
identical with its specific gravity as in the CG.S. 
system. 

Even those who may have a preference for some other 
units will nevertheless admit the advantage of having a 
variety of results, from various branches of physics, re- 
duced from their original multiplicity and presented in 
one common scale. 

22. The adoption of one common scale for all quan- 
tities involves the frequent use of very large and very 



24 UNITS AND PHYSICAL CONSTANTS, [chap. ii. 

small numbers. Such numbers are most conveniently 
written by expressing them as the product of two factors, 
one of which is a power of 10 ; and it is usually advan- 
tageous to effect the resolution in siich a way that the 
exponent of the power of 10 shall be the characteristic of 
the logarithm of the number. Thus 3240000000 will 
be written 3-24 x 10», and -00000324 will be written 
3-24 X 10-«. 



26 



CHAPTER III. 

MECHANICAL UNITS. 

Value of g. 

23. AccELEBATiON is defined as the rate of increase of 
velocity per unit of time. The C.G.S. unit of accelera- 
tion is the acceleration of a body whose velocity increases 
in every second by the C.G.S. unit of velocity — namely, 
by a centimetre per second. The apparent acceleration 
of a body falling freely under the action of gravity in 
vacuo is denoted by g. The value of ^r in C.G.S. units 
at any part of the earth's surface is approximately given 
by the following formula, 

g = 980-6056 - 2-5028 cos 2 A - -OOOOOSA, 

A. denoting the latitude^ and h the height of the station 
(in centimetres) above sea-level. 

The constants in this formula have been deduced from 
numerous pendulum experiments in different localities, 
the length I of the seconds' pendulum being connected 
with the value of g by the formula g = ttH, 

Dividing the above equation by tt^ we have, for the 
length of the seconds' pendulum, in centimetres, 

/= 99-3562 - -2536 cos 2 A - -OOOOOOSA. 



26 



UNITS AND PHYSICAL CONSTANTS. [chap. 



At sea-level these formulse give the following values for 
the places specified : — 





Latitude. 


Value of g. 


Value of U 


« 

Equator, - 


6 


978 10 


99-103 


Latitude 45°, - 


45 


980*61 


99*356 


Munich, - 


48 9 


980-88 


99-384 


Paris, - 


48 50 


980-94 


99-390 


Greenwich, 


61 29 


981-17 


99-413 


Gottingen, 


51 32 


981-17 


99-414 


Berlin, 


52 30 


981-25 


99-422 


Dublin, - 


53 21 


981-32 


99-429 


Manchester, 


53 29 


981-34 


99-430 


Belfast, - 


54 36 


981-43 


99-440 


Edinburgh, 


55 57 


981-54 


99-451 


Aberdeen, 


57 9 


981-64 


99-461 


Pole, 


90 


983-11 


99-610 



The difference between the greatest and least values 
(in the case of both g and I) is about —— of the mean 



196 



value. 



24. The Standards Department of the Board of Trade, 
being concerned only with relative determinations, has 
adopted the formula 

(7 = 5r,(l--00257cos2A)A-|A\ 

k denoting the latitude, h the height above sea-level, K 
the earth's radius, g^ the value of g in latitude 45 ** at sea- 
level, which may be treated as an unknown constant 
multiplier. Putting for It its value in centimetres, the 
formula gives 

g^g^{\ - -00257 cos 2A - 1-96A x lO"*), 

where h denotes the height in centimetres. 



in.] MECHANICAL UNITS. 27 

The formula which we employed in the preceding 
section gives 

^=^,(1 - -00255 cos 2A)(l - ^V 
As regards the factor dependent on height, theory indi- 
cates 1 - ~ as its correct value for such a case as that of 

Xv 

a balloon in mid-air over a low-lying country ; the value 
1 - J :g- may be accepted as more correct for an elevated 

plateau on the earth's surface. 

Force, 

25. The C.G.S. unit of force is called the dyne. It is 
the force which, acting upon a gramme for a second, 
generates a velocity of a centimetre per second. 

It may otherwise be defined as the force which, acting 
upon a gramme, produces the C.G.S. unit of acceleration, 
or as the force which, acting upon any mass for 1 second, 
produces the C.G.S. unit of momentum. 

To show the equivalence of these three definitions, let 
m denote mass in grammes, v velocity in centimetres per 
second, t time in seconds, F force in dynes. 

Then, by the second law of motion, we have 

1 .. force 
acceleration = ; 

mass 

P 
that is, if a denote acceleration in C.G.S. units, a= - ; 

m 

hence, when a and m are each unity, F will be unity. 

Again, by the nature of uniform acceleration, we have 

v-at, V denoting the velocity due to the acceleration a, 

continuing for time t. 



28 UNITS AND PHYSICAL CONSTANTS. [cfHAP. 

Hence we have F = ma = — . Therefore, if mv = 1 

t 

and t=ly we have F= 1. 

As a particular case, if in -I, v = l, t = l, we have 
F = l. 

26. The force represented by the weight of a gramme 
varies from place to place. It is the force required to 
sustain a gramme in vacuo, and would be nil at the 
earth's centre, where gravity is nil. To compute its 
amount in dynes at any place where g is known, observe 
that a mass of 1 gramme falls in vacuo with acceleration 
g. The force producing this acceleration (namely, the 
weight of the gramme) must be equal to the product of 
the mass and acceleration, that is, to g. 

The weight (when weight means force) of 1 gramme is 
therefore g dynes ; and the weight of m grammes is mg 
dynes. 

27. Force is said to be expressed in gravitation-measure 
when it is expressed as equal to the weight of a given 
mass. Such specification is inexact unless the value of 
g is also given. For purposes of accuracy it must always 
be remembered that the pound, the gramme, etc., are, 
strictly speaking, units of mass. Such an expression as 
" a force of 100 tons " must be understood as an abbrevia- 
tion for " a force equal to the weight [at the locality in 
question] of 100 tons." 

28. The name poundal has recently been given to the 
unit force based on the pound, foot, and second ; that is, 
the force which, acting on a pound for a second, gene- 
rates a velocity of a foot per second. It is — of the 



HI.] MECHANICAL UNITS. 29 

weight of a pound, g denoting the acceleration due to 
gravity expressed in foot-second units, which is about 
32*2 in Great Britain. 

To compare the poundal with the dyne, let x denote 
the number of dynes in a poundal ; then we have 

gm. cm. __ lb. ft. 



X 



sec. 2 sec. 2 



x^— , — = 453-59 X 30-4797 = 13825. 



gm. cm. 



Work mui Energy, 

29. The C.G.S. unit of work is called the erg. It is 
the amount of work done by a dyne working through a 
distance of a centimetre. 

The C.G.S. unit of energy is also the erg, energy being 
measured by the amount of work which it represents. 

30. To establish a rule for computing the kinetic energy 
(or energy due to the motion) of a given mass moving with 
a given velocity, it is sufficient to consider the case of 
a body falling in vacuo. 

When a body of m grammes falls through a height of /* 
centimetres, the working force is the weight of the body 
— ^that is, gTTh dynes, which, multiplied by the distance 
worked through, gives gmh ergs as the work done. But 
the velocity acquired is such that «?- = 2^A. Hence we 
have gmh = ^v^. 

The kinetic energy of a mass of m grammes moving 
with a velocity of v centimetres per second is therefore 
i^v^ ergs ; that is to say, this is the amount of work 
which would be required to generate the motion of 
the body, or is the amount of work which the body 



30 UNITS AND PHYSICAL CONSTANTS. [chap. 

would do against opposing forces before it would come 
to rest. 

31. Work, like force, is often expressed in gravitation- 
measv/re. Gravitation units of work, such as the foot- 
pound and kilogramme-metre, vary with locality, being 
proportional to the value of g. 

One gramme-centimetre is equal to g ergs. 
One kilogramme-metre is equal to 100,000 g ergs. 
One foot-poundal is 463-69 x (30-4797)^ = 421390 ergs. 
One foot-pound is 13,825 g ergs, which, if g be taken 
as 981, is 1-356x10" ergs. 

32. The C.G.S. unit rate of working is 1 erg per second. 
Watt's " horse-power " is defined as 560 foot-pounds per 
second. This is 7*46 x 10^ ergs per second. The ** force de 
cheval " is defined as 75 kilogrammetres per second. This 
is 7*36 X 10^ ergs per second. We here assume ^ = 981. 

A new unit of rate of working has been lately intro- 
duced for convenience in certain electrical calculations. 
It is called the Watt^ and is defined as 10*^ ergs per second. 
A thousand watts make a kilowatt The following 
tabular statement will be useful for reference. 

1 Watt = 10*^ ergs per second = -00134 horse-power 
= '737 foot-pounds per second = -101 9 kilogram- 
metres per second. 
1 Kilowatt = 1 -34 horse-power. 

1 Horse-power = 550 foot-pounds per second = 76*0 
kilogrammetres per second = 746 watts = 1 -01 385 
force de cheval. 
1 Force de cheval = 75 kilogrammetres per second 
= 642 -48 foot-pounds per second =736 watts 
= -9863 horse-power. 



ni.l MECHANICAL UNITS. 31 

Examples, 

1. If a spring balance is graduated so as to show the 
masses of bodies in pounds or grammes when used at the 
equator, what will be its error when used at the poles, 
neglecting effects of temperature ? 

Ana, Its indications will be too high by about y^^ of 

the total weight. 

2. A cannon-ball, of 10,000 grammes, is discharged 
with a velocity of 45,000 centims. per second. Find its 
kinetic energy. 

Ana, I X 10000 x (46000)* = 1*0125 x lO^^ ergs. 

3. In last question find the mean force exerted upon 
the ball by the powder, the length of the barrel being 
200 centims. 

Ana. 5-0625 x lO^^ dynes. 

4. Given that 42 million ergs are equivalent to 1 
gramme-degree of heat, and that a gramme of lead at 
10° 0. requires 15*6 gramme-degrees of heat to melt it; 
find the velocity with which a leaden bullet must strike a 
target that it may just be melted by the collision, suppos- 
ing all the mechanical energy of the motion to be converted 
into heat and to be taken up by the bullet. 

We have Jt;*=15'6xJ, where J = 42x10^. Hence 
»*=1310 millions; i?-36'2 thousand centims. per 
second. 

5. With what velocity must a stone be thrown verti- 
cally upwards at a place where ^ is 981 that it may rise 
tea height of 3000 centims.? and to what height would 
it ascend if projected vertically with this velocity at the 
surface of the moon, where ^ is 150 ? 

Ana, 2426 centims. per second ; 19620 centims. 



32 UNITS AND PHYSICAL CONSTANTS. [chap. 

Centrifugal Force, 

33. A body moving in a curve must be regarded as 

continually falling away from a tangent The accelera- 

v 
tion with which it falls away is — , i? denoting its velocity 

and r the radius of curvature. The acceleration of a 
body in any direction is always due to force urging it in 
that direction, this force being equal to the product of 
mass and acceleration. Hence the normal force on a body 
of m grammes moving in a curve of radius r centimetres, 

with velocity v centimetres per second, is dynes. This 

force is directed towards the centre of curvature. The 
equal and opposite force with which the body reacts is 
called centrifugal force. 

If the body moves uniformly in a circle, the time 

27n- 
of revolution being T seconds, we have v = ~ifr'i 

v^ /27r\2 
hence "^("rfrj^j aiid the force acting on the body is 

/27r\2 
//*n rn-j dynes. 

If n revolutions are made per minute, the value of T is 



— , and the force is mr 



('Sj ^^^ 



Examples. 

1. A body of m grammes moves uniformly in a circle 
of radius 80 centims., the time of revolution being ^ of a 



in.] MECHANICAL UNITS. 33 

second. Find the centi'ifugal force, and compare it with 
the weight of the body. 

-^1 X 80 = m X 647r* 

X 80 = 50532 m dynes. 

The weight of the body (at a place where ^ is 981) is 
981 m dynes. Hence the centrifugal force is about b2\ 
times the weight of the body. 

2. At a bend in a river, the velocity in a certain part 
of the surface is 170 centims. per second, and the radius 
of curvature of the lines of flow is 9100 centims. Find 
the slope of the surface in a section transverse to the lines 
of flow. 

Ana, Here the centrifugal force for a gramme of the 
water is 0^=3-176 dynes. If ^ be 981 the slope will 

be = -r^ ; that is, the surface will slope upwards 

981 o09 

from the concave side at a gradient of 1 in 309. The 

general rule applicable to questions of this kind is that 

the resultant of centrifugal force and gravity must be 

normal to the surface. 

3. An open vessel of liquid is made to rotate rapidly 
round a vertical axis. Find the number of revolutions 
that must be made per minute in order to obtain a slope 
of 30° at a part of the surface distant 10 centims. from 
the axis, the value of ^ being 981. 

f 
Ans, We must have tan 30° = ^, where /* denotes the 

9 
intensity of centrifugal force — that is, the centrifugal force 

per unit mass. We have therefore 

c 



34 UNITS AKD PHYSICAL CONSTANTS. [chap. 



981 tan 30° = lo(|^Y' ** **«'^°*»°8 



w denoting the number of 
revolutions per minute, 



o o 



90 
Hence n = 71'9. 



4. For the intensity of centrifugal force at the equator 
due to the earth's rotation, we have r = earth's radius 
= 6*38 X 10^, T = 86164, being the number of seconds in 
a sidereal day. 



•■• ^'"-i^i)-^-^^- 



This is about ^— of the value of g. 

If the earth were at rest, the value of g at the equator 
would be greater than at present by this amount. If the 
earth were revolving about 1 7 times as fast as at present, 
the value of g at the equator would be nil. 

Supplemental Section. 

On the help to he derived from Dimensions in investi- 
gating Physical Formulce, 

When one physical quantity is known to vary as some 
power of another physical quantity, it is often possible to 
find the exponent of this power by reasoning based on 
dimensions, and thus to anticipate the results — or some 
of the results — of a dynamical investigation. 

Examples, 

1. The time of vibration of a simple pendulum in a 
small arc depends on the length of the pendulum and the 
intensity of gravity. If we assume it to vary as the wi* 



HL] MECHANICAL UNITS. 35 

power of the length, and as the ri'* power of g, and to be 
independent of everything else, the dimensions of a time 
must equal the m** power of a length, multiplied by the w'* 
power of an acceleration, that is 

T = L"*(LT-2)" = L*" L** T-2" 

__ T »» + n T^-2n 

Since the dimensions of both members are to be identical, 
we have, by equating the exponents of T, 

1 = - 2^1, whence n= - ^, 

and by equating the exponents of L, 

m + n = 0, whence m = J ; 

that is, the time of vibration varies directly as the square 
root of the length, and inversely as the square root of ^. 

2. The velocity of sound in a gas depends only on the 
density D of the gas and its coefficient of elasticity E, and 
we shall assume it to vary as D"* E". 

The dimensions of velocity are LT"\ 

The dimensions of density, or — = , are ML~'. 

volume 

The dimensions of E, which will be explained in the 

chapter on stress and strain, are , or (MLT"^)L~^, or 

area 

ML-i T-l 
The equation of dimensions is 

_. IJTm+n T —3m—n rp— 2n 

whence, by equating coefficients, we have the three 
equations 

1 = - 3m - 71, - 1 = - 2n, m + n = 0, 
to determine the two unknowns m and n. 



36 UNITS AND PHYSICAL CONSTANTS. [c 

The second equation gives at once 

w = J. 
The third then gives 

m= -J, 

and these values will be found to satisfy the first equs 

also. 

The velocity, then, varies directly as the square ro< 

E, and inversely as the square root of D. 

3. The frequency of vibration f for a musical st 
(that is, the number of vibrations per unit time) dep 
on its length I, its mass niy and the force with whic 
is stretched F. 

The dimensions of/ are T"\ 

„ „ X „ XU.XJJ. . 

Assume that/ varies as I'm^ F'. Then we have 

T-^ = L*M»'M^L*T-^ 

= L'+'M»'+*T-% 
giving -l=-20, x + z = 0, y + z=^0; 

whence ^ = h ^= "h 1/-"^' 



Hence /varies as ^ /.;— . 



4. The angular acceleration of a uniform disc roun( 
axis depends on the applied couple G, the mass of the 
M, and its radius E. 

Assume it to vary as G* M*' R*. 

The dimensions of angular acceleration are T"-^. 

G „ ML^T- 

Hence we have 

T-2 = M* W T-^ M*' L\ 



irr.] MECHANICAL UNITS. 37 

gi^v^ing "2= -2xj re + 1/ = 0, 2x + z = 0, 
w-lience 05 = 1, y=— 1, «=-2. 

Hence the angular acceleration varies as . 

In the following example the information obtained is 
less complete : — 

5. The range of a projectile on a horizontal plane 
through the point of projection depends on the initial 
velocity V, the intensity of gravity g, and the angle of 
elevation cu 

The dimensions of range are L. 

T.T-2 

» »» 



» )> 



9 » LT- 

a „ L^ T^, and the dimensions 
of all powers of a are L^ T^. Hence we can draw no 
inferences as to the manner in which a enters the expres- 
sion for the range. The dimensions of this expression will 
depend upon V and g alone. 
Assume that the range varies as Y"*^". Then 

L = (LT-i)'»(LT-% 

__ T m+n nn— m— 2n . 

giving m + n=ly m+2n = 0, 

whence m = 2, w = - 1. 

. Y2 . . 

Hence the range varies as — when a is given. 



38 



CHAPTER IV. 



HYDROSTATICS. 



34. The following table of the relative density of water 
at various temperatures (under atmospheric pressure), iihe 
density at 4° C. being taken as unity^ is from Eofisetti's 
results deduced from all the best experiments (Ann. Gb. 
Phys. X. 461 ; xvii. 370, 1869) :— 



Temp. 
Cent. 


Relative 


Temp. 


Relative 


Temp. 


Relative 


Density. 


Cent. 


Density. 


Cent. 


Density. 


o 




•999871 




13 


•999430 


o 

35 


•99418 


1 


•999928 


14 


•999299 


40 


•99235 


2 


•999969 


15 


•999160 


45 


•99037 


3 


•999991 


16 


•999002 


50 


•98820 


4 


i-oooooo 


17 


•998841 


55 


•98582 


5 


•999990 


18 


•998654 


60 


•98338 


6 


•999970 


19 


•998460 


65 


•98074 


7 


•999933 


20 


•998259 


70 


•97794 


8 


•999886 


22 


•997826 


76 


•97498 


9 


•999824 


24 


•997367 


80 


•97194 


10 


•999747 


26 


•996866 


85 


•96879 


11 


•999655 


28 


•996331 


90 


•96656 


12 


•999549 


30 


•995765 


100 


•95865 



35. According to Kupffer's observations, as reduced 
by Professor W. H. Miller, the absolute density (in 
grammes per cubic centimetre) at 4° is not 1, but 
1 •00001 3. Multiplying the above numbers by this 






01£AP. IV. 


1 


HYDRC 


)STATICS. 




39 


factor, we obtain 


the following table of absolute den- 


si-ties : — 








lemp i Density. 


Temp. 


Density. 


Temp. 


Density. 







'999884 


o 

13 


•999443 1 


o 

35 


•99469 


1 


•999941 


14 


•999312 


40 


•99236 


1 2 


•999982 


15 


•999173 . 


45 


•99038 


3 


1000004 


16 


•999015 1 


50 


•98821 


4 


1000013 


17 


•998854 


55 


•98583 


5 


1-000003 


18 


•998667 


1 60' 


•98339 


6 


•999983 


19 


•998473 


65 


•98075 


7 


•999946 


20 


•998272 


70 


•97795 


8 


•999899 


22 


•997839 


75 


•97499 


9 


•999887 


24 


•997380 


80 


•97195 


10 


•999760 


26 


•996879 


85 


•96880 


11 


•999668 


28 


•996344 


90 


•96557 


12 


•999562 


30 


•995778 


100 


•95866 



36. The volume, at temperature t\ of the water which 
<H5cupies unit volume at 4°, is approximately 

1 + A(^ - 4)2 - B{t - 4)2« 4- C{t - 4)3, 
^here 

A = 8^38 X lO-«, 
B = 3-79 X 10-^ 
C = 2^24 X 10-8; 

^•xid the relative density at temperature ^° is given by the 
*«tnie formula with the signs of A, B, and C reversed. 
The rate of expansion at temperature t° is 

2A(«-4)-2^6B(«-4)^« x 3C{t-4:)\ 
In determining the signs of the terms with the frac- 
tional exponents 2^6 and 1^6, these exponents are to be 
regarded as odd. 

37. The following Table of Densities has been compiled 
by collating the best authorities, but is only to be taken 



40 



UNITS AND PHYSICAL CONSTANTS. [chap. 



aa giving rough approximations. Most of the densities 
vary between wide limits in different specimens : — 



Solids. 



Aluminium, 
Antimony, .. 
Bismuth, .... 

Brass, 

Copper, 

Gold, 

Iron, 

Lead, 

Nickel, 

Platinum, ... 

Silver, 

Sodium, 

Tin, 

Zinc, 

Cork 

Oak, 

Ebony, 

Ice, 



1 



2-6 

6-7 

9-8 

8-4 

8-9 
19-3 

7-8 
11-3 

8-9 
21-5 
10-5 
•98 

7-3 

7-1 
•24 

•7 to 1 

•1 to 1-2 

•918 



>» 



It 



Carbon (diamond),., 
(graphite),., 
(gas carbon), 
,, (wood charcoal), 
Phosphorus (ordi- 
nary), 

,, (red),... 

Sulphur (roll), 

Quartz (rock cry- 
stal), 

Sand (dry), 

Clay, 

Brick, 

Basalt, 

Chalk, 1" 

Glass (crown), 2' 

„ (flint) 3" 

Porcelain, 



3 
2 
I 
I 



5 
3 
9 
6 



183 

2-2 

2^0 



2-65 

142 

1-9 

21 

3-0 

8 to 2-8 
5 to 2^7 
to 3-5 

2-4 



Liquids at 0° C. 



Sea water, 1'026 

Alcohol, '8 

Chloroform, 1*5 

Ether, ^73 

Bisulphide of Carbon, . . 1 "29 

Glycerine, 1^27 

Mercury, 13*596 



Sulphuric Acid,... . 

Nitric Acid, 

Hydrochloric Acid, 

Milk, 

Oil of Turpentine,.. 
Linseed, 



it 



I 85 
I^56 
I 27 
1-03 
•87 
•94 



Mineral, •76to-83 

More exactly, the density of mercury at 0° C, as com- 
pared with water at the temperature of maximum density, 
under atmospheric pressure, is 13^5956. 

38. If a body weighs m grammes in vacuo and m' 
grammes in water of density unity, the volume of the 
body is w - m' cubic centims. ; for the mass of the water 
displaced is m-m/ grammes, and each gramme of this 
water occupies a cubic centimetre. 



IV.] HYDROSTATICS. 41 

Examples, 

1. A glass cylinder, I centims. long, weighs m grammes 
in vacuo and m! grammes in water of unit density. Find 
its radius. 

Solution. Its section is Trr^, and is also — - — ; hence 

V 

2 _ m - m' 

"^ — vr' 

2. Find the capacity at 0° C. of a bulb which holds m 
grammes of mercury at that temperature. 

Solution, The specific gravity of mercury at 0° being 
13*5956 as compared with water at the temperature of 
maximum density, it follows that the mass of 1 cubic 
centim. of mercury is 13-5956 x 1-000013 = 13-5958, say 

13-596. Hence the required capacity is cubic 

lo *o»/0 

^^ntims. 

3. Find the total pressure on a surface whose area is A 
square centims. when its centre of gravity is immei-sed to 
* depth of h centims. in water of unity density, atmos- 
pheric pressure being neglected. 

Ana, Ah grammes weight ; that is, gAh dynes. 

4. If mercury of specific gravity 13*596 is substituted 
fcr water in the preceding question, find the pressure. 

Ans. 13-596 Ah grammes weight; that is, 13-596 gAh 

5. If h be 76, and A be unity in example 4, the answer 
becomes 1033*3 grammes weight, or 1033-3^^ dynes. 

For Paris, where g is 980*94, this is 1*0136 xlO« 
dynes. 



42 UNITS AND PHYSICAL CONSTANTS. [chap. 

Barometric Pressure. 

39. The C.G.S. unit of pressure intensity (that is, of 
pressure per unit area) is the pressure of a dyne per 
square centim. 

At the depth of h centims. in a uniform liquid whose 
density is d [grammes per cubic centim.], the pressure due 
to the weight of the liquid is ghd dynes per square centim. 

The pressure-intensity due to the weight of a column of 
mercuiy at 0° C, 76 centims. high, is found by putting 
A =76, c?= 13-596, and is 1033%. It is therefore 
different at different localities. At Paris, where g is 
980-94, it is 1-0136 x 10«; that is, rather more than a 
megadyne* per square centim. To exert a pressure of 
exactly one megadyne per square centim^^ the height of 
the column at Paris must be 74*98 centims. 

At Greenwich, where g is 981-17, the pressure due to 
76 centims. of mercury at 0** C. is 1*0138 x 10^; and the 
height which would give a pressure of 10^ is 74*964 
centims.^ or 29*514 inches. 

Convenience of calculation would be promoted by 
adopting the pressure of a megadyne per square centim., 
or 10^ C.G.S. units of pressure-intensity, as the standard 
atmosphere. 

The standard now commonly adopted (whether 76 
centims. or 30 inches) denotes different pressures at 
different places, the pressure denoted by it being pro- 
portional to the value of g. 

We shall adopt the megadyne per square centim. as 
our standard atmosphere in the present work. 

*The prefix mega denotes multiplication by a million. A 
megadyne is a force of a million dynes. 



IV.] HYDROSTATICS. 4a 

Examples, 

1. What must be the height of a column of water of 
unit density to exert a pressure of a megadyno per square 
centim. at a place where g is 981 ? 

Ana. ^^^^ = 1019-4 centims. This is 33-445 feet. 

2. What is the pressure due to an inch of mercury at 
0* C. at a place where g is 981 ? (An inch is 2*54 
centims.) 

Am, 981 X 2-54 x 13*596 = 33878 dynes per square 
centim. 

3. What is the pressure due to a centim. of mercury at 
0° C. at the same locality ? 

Am. 981x13-596 = 13338. 

4. What is the pressure due to a kilometre of sea-water 
of density 1-027, g being 981 1 

Ans, 981 X 105 x 1-027 = 1*0075 x 10^ dynes per square 
centim., or 1-0075 x 10^ megadynes per square centim.; 
that is, about 100 atmospheres. 

5. What is the pressure due to a mile of the same 
water 1 

Ana, 1*6214 x 10^ C.G.S. units, or 162*14 atmospheres 
[of a megadyne per square centim.]. 

Density of Air, 

40. Regnault found that at Paris, under the pressure 
of a column of mercury at 0°, of the height of 76 centims., 
the density of perfectly dry air was -0012932 gramme per 
cubic centim. The pressure corresponding to this height 
of the barometer at Paris is 1*0136 x 10^ dynes per square 



44 



UNITS AND PHYSICAL CONSTANTS. [chap- 



centiui. Hence, by Boyle's law, we can compute the 
density of dry air at 0® C. at any given pressure. 

At a pressure of a megadyne (10® dynes) per square 

centim. the density will be— -— — ^= '0012759. 

The density of dry air at 0** C. at any pressure p (dynes 
per square centim.) is 

;?xl-2759xlO-» . . . . W 

Uocample. 

Find the density of dry air at 0° C, at Edinburgb, 
under the pressure of a column of mercury at 0° C, of 
the height of 76 centims. 

Here we have p = 981-54 x 76 x 13-596 = 1*0142 x W. 

Am. Required density = 1-2940 x lO"* = -0012940 
gramme per cubic centim. 

41. Absolute Densities of Gases, in grammes per cubic 
centim. y at 0° C, and a pressure q/* 10® dynes per 
square centim,. 

Mass of a cubic Volume of a g^ramme 
centim. in grammes, in cubic centims. 

0012759 783-8 

0014107 708-9 

0012393 806-9 

00008837 11316-0 

0019509 512-6 

0012179 821-1 

0007173 1394-1 

0030909 323-5 

0019433 514-6 

0013254 754-5 

0026990 370-5 

0022990 435-0 

0012529 798-1 

0007594 1316-8 



Air, dry, 

Oxygen, 

Nitrogen, 

Hydrogen, 

Carbonic Acid, 

. „ Oxide, 

Marsh Gas, 

Chlorine, 

Protoxide of Nitrogen,.. 
Binoxide ,, 

Sulphurous Acid, 

Cyanogen, 

defiant Gas, 

Ammonia, 



,] HYDROSTATICS. 45 

The numbers in the second column are the reciprocals 

those in the first. 

The numbers in the first column are identical with the 
*X^6<5ific gravities referred to water as unity. 

Assuming that the densities of gases at given pressure 
^-^=id temperature are directly as their atomic weights, we 
*-^^ve for any gas at zero 

j9v/A=M316xl0i0m; 
^^ denoting its volume in cubic centims., m its mass in 

immes, p its pressure in dynes per square centim., and 

its atomic weight referred to that of hydrogen as unity. 

Height of HoTnogeneous Atmosphere, 
42. We have seen that the intensity of pressure at 
^pth A, in a fluid of uniform density dy is ghd when the 
Measure at the upper surface of the fluid is zero. 
The atmosphere is not a fluid of uniform density ; but 
is often convenient to have a name to denote a height 
such that p = ^HD, where p denotes the pressure and 
the density of the air at a given point. 
It may be defined as the height of a column of uniform 
Qidd having the same density as the air at the pointy 
'^hich would exert a pressure equal to that existing at 
"^te point. 

If the pressure be equal to that exerted by a column of 
^Xiercury of density 13 '5 9 6 and height A, we have 

p = ghyi 13-596; 

.-. HD = A X 13-596, H = ^^ ^ 13-596 

If it were possible for the whole body of air above the 
point to be reduced by vertical compression to the density 
which the air has at the point, the height from the point 



46 UNITS AND PHYSICAL CONSTANTS. [chap. 

up to the summit of this compressed atmosphere would be 
equal to H, subject to a small correction for the variation 
of gravity with height. 

H is called the height of the homogeneous atmosphere at 
the point considered. Pressure-height would be a better 
name. 

The general formula for it is 

H=^; ... (5) 

and this formula will be applicable to any other gas as 
well as dry air, if we make D denote the density of the 
gas (in grammes per cubic centim.) at pressure p. 

If, instead of jp being given directly in dynes per square 
centim., we have given the height A of a column of liquid 
of density d which would exert an equal pressure, the 
formula reduces to 

H = '^. (6) 

43. The value of ^ in formula (5) depends only on the 

nature of the gas and on the temperature ; hence, for a 
given gas at a given temperature, H varies inversely 
•as g only. 

For dry air at zero we have, by formula (4), 

^ 7-8376 X 108 

xl = . 

9 
At Paris, where g is 980*94, we find 

H = 7-990x105. 

At Greenwich, where g is 981*17, 

H = 7-988 xlO^ 



IV.] HYDROSTATICS. 47 

Examples, 

1. Find the height of the homogeneous atmosphere at 
Paris for dry air at 10° C, and also at 100° C. 

Ans. For given density, p varies as 1 x -00366 t, t de- 
noting the temperature on the Centigrade scale. Hence 
we have, at 10° C, 

H = 1-0366 X 7-99 x 10^ = 8*2825 x 10^ ; 
and at 100° C, 

H = 1-366 X 7-99 x 105 = 1-0914 x 10». 

2. Find the height of the homogeneous atmosphere for 
hydrogen at 0°, at a place where g is 981. 

Here we have 

K=P- = — = 1-1535x107. 

gd 981x8-837x10-'^ 

JDiminution of Density with increase of Height in the 

Atmosphere. 

44. Neglecting the variation of gravity with height, 
the variation of H as we ascend in the atmosphere would 
depend only on variation of temperature. In an atmos- 
phere of uniform temperature H will be the same at all 
heights. In such an atmosphere, an ascent of 1 centim. 
will involve a diminution of the pressure (and therefore 

of the density) by — of itself, since the layer of air which 

H 

has been traversed is =_. of the whole mass of superincum- 

H 

bent air. The density therefore diminishes by the same 
fraction of itself for every centim. that we ascend; in 
other words, the density and pressure diminish in geo- 
metrical progression as the height increases in arithmetical 
progression. 



48 UNITS AND PHYSICAL CONSTANTS. [chap. 

Denote height above a fixed level by 05, and pressure 

by p. Then, in the notation of the differential calculus, 

, dx dp 

we nave ._- = - ^ 

±1 p 

and if jt?i, pg ^^ *^® pressures at the heights a^, ajg* ^^ 
deduce 

x.-x^ = ll log,?! = H X 2-3026 log^o^^- • • (7) 

In the barometric determination of heights it is usual 
to compute H by assuming a temperature which is the 
arithmetical mean of the temperatures at the two heights. 

For the latitude of Greenwich formula (7) becomes 

(Tg - a?! = (1 X -00366 1) 7-988 x 10^ x 2*3026 log^ 

P2 

= (lx -00366 1,839,300 log^i, • • (8) 

P2 
t denoting the mean temperature, and the logarithms 

being common logarithms. 

To find the height at which the density would be halved, 

variations of temperature being neglected, we must put 2 

for O in these formulae. The required height will be H 

P2 
log, 2, or, in the latitude of Greenwich, for temperature 

O"* C, will be 

1-8393 X 106 X -30103 = 553700. 
The value of log, 2, or 2-3026 log^o 2, is 

2-3026 X -30103 =-69315. 
Hence for an atmosphere of any gas at uniform tempera- 
ture, the height at which the density would be halved is 
the height of the homogeneous atmosphere for that gas, 
multiplied by -69315. The gas is assumed to obey 
Boyle's law. 



IV.] 



HYDROSTATICS. 



49 



Examples. 

1. Show that if the pressure of the gas at the lower 
station and the value of g be given, the height at which 
the density will be halved varies inversely as the density. 

2. At what height, in an atmosphere of hydrogen at 
0** C, would the density be halved, g being 981 1 

Ana. 7-9954 xlO« 

45. The phenomena of capillarity, soap-bubbles, etc., 
can be reduced to quantitative expression by assuming a 
tendency in the surface of every liquid to contract. The 
following table exhibits the intensity of this contractile 
force for various liquids at the temperature of 20** C. 
The contractile force diminishes as the temperature in- 
creases. 

Sttperficial tensions at 20° C, in dynes per linear centim.^ 
dediiced from Quincke* s results. 





Density. 


Tension of Surface separating 
the Liquid from 


Air. 


Water. 


Mercury. 


Water, 

Mercury, - - - 
BiBulphide of Carbon, - 
Chloroform, - 
Alcohol, 
OUveOil, - 
Turpentine, - 
Petroleum, - 
Hydrochloric Acid, 
Solution of Hyposul- ) 
phite of Soda, - - j 


0-9982 
13-5432 

1-2687 

1-4878 
•7906 
•9136 
•8867 
-7977 

11 

11248 


81 
540 
32 1 
30-6 
25-5 
36-9 
29-7 
31-7 
70-1 

77-5 



418 
41-75 
29-5 

26'-66 
11-55 

27-8 

• • • 

• • • 


418 

372-5 
399 
399 
335 
250-5 
284 
377 

442-5 



The values here given for water and mercury are only 
i^plicable when special precautions are taken to ensure 

D 



50 



UNITS AND PHYSICAL CONSTANTS. [chap. 



cleanliness and purity. Without such precautions smaller 
values will be obtained. (Quincke in Wied. Ann,, 1886, 
page 219.) 

The following values are from the observations of A. M. 
Worthington (Proc, Roy, Soc, June 16, 1881), at tempera- 
tures from 15** to 18° C, for surfaces exposed to air : — 



^Surface Tension, 

In gm. per cm. 

Water, -072 to -080 

Alcohol, -02586 

Turpentine, -02818 

Olive Oil, 03373 

Chloroform, -03025 



In dynes per cm. 

70-6 to 78-5 
25-3 
27-6 
33-1 
29-6 



46. Very elaborate measurements of the thicknesses of 
soap films have been made by Reinold and Riicker (Phil, 
Trans,, 1881, p. 456; and 1883, p. 651). When so thin 
as to appear black, the thickness varied from 7 '2 to 14-6 
millionths of a millimetre, the mean being 11-7. This is 
1-17 X 10~® centimetra The following thicknesses were 
observed for the colours of the successive orders : — 



Thickness, 
cm. 

First Order— 

Red, 2-84x10-5 

Second Order — 

Violet, 3-05 

Blue, 3-53 

Green, 4-09 

Yellow, 4-54 

Orange, 4-91 

Red, 5-22 

Third Order— 

Purple, 5-59 

Blue, 5-77 

„ 603 

Green, 6-66 









Thickness, 
cm. 

Yellow, 7.10x10-'^ 

Red, 7-65 

Bluish Red, 8-15 



>» 



Fourth Order— 

Green, 8-41 

„ 8-93 

Yellow-Green, . . 9-64 
Red, 10-52 



»» 



>» 
»» 
»> 
»» 



ft 



Fifth Order — 

Green, 1-119x10-* 

„ M88 „ 

Red, 1-260 ,, 

„ 1-336. 



>» 



IV.] 



HYDROSTATIO=I. 



51 



Sixth Order — 
Green, 



)» 



Red, 

>» 

Sevbnth Obdeu- 
Green, 



Thickness, 
cm. 

I -410 X 10-* 
1-479 „ 
1-548 „ 
1-627 „ 



Green, 
Red,... 



>» 



Thickness, 
cm. 

1-787x10-* 

1-869 

1-936 



>> 



1-705 



Eighth Order- 

Green, 

Red, 



»» 



2004 „ 
2115 „ 



») 



46a. Depression of the barometrical column due to 
capillarity, according to Pouillet : — 



Internal 




Internal 




Internal 




Diameter 


Depression. 


Diameter 


Depression. 


Diameter 


Depression. 


of tube. 




of tube. 




of tube. 




inm. 


mm. 


mm. 


nun. 


mm. 


mm. 


2 


4-579 


8-5 


•604 


15 


•127 


2-6 


3-595 


9 


•534 


15-5 


-112 


3 


2-902 


9-5 


•473 


16 


-099 


3-5 


2-415 


10 


•419 


16-0 


•087 


4 


2-053 


10-5 


•372 


17 


-077 


4-5 


1-752 


11 


•330 


17 5 


•068 


,5 


1-507 


11-5 


•293 


18 


•060 


5-6 


1-306 


12 


•260 


18^5 


-053 


6 


1-136 


12-5 


-230 


19 


•047 


6-5 


•995 


13 


•204 


195 


•041 


7 


-877 


13-5 


-181 


20 


-036 


7-5 


-775 


14 


•161 


205 


-032 


8 


•684 


14-5 


-143 


21 


-028 



52 



CHAPTER V. 

STRESS, STRAIN, AND RESILIENCE. 

47. In the nomenclature introduced hy Kankine, and 
adopted hy Thomson and Taifc, any change in the shape 
or size of a body is called a strain, and an action of force 
tending to produce a strain is called a stress. We shall 
always suppose strains to be small ; that is, we shall sup- 
pose the ratio of the initial to the final length of every 
line in the strained body to be nearly a ratio of equality. 

48. A strain changes every small spherical portion of 
che body into an ellipsoid ; and the strain is said to be 
homogeneous when equal spherical portions in all parts 
of the body are changed into equal ellipsoids with their 
corresponding axes equal and parallel. When the strain 
consists in change of volume, unaccompanied by change 
of shape, the ellipsoids are spheres. 

When strain is not homogeneous, but varies continu- 
ously from point to point, the strain at any point is 
defined by attending to the change which takes place 
in a very small sphere or cube having the point at its 
centre, so small that the strain throughout it may be 
regarded as homogeneous. In what follows we shall 
suppose strain to be homogeneous, unless the contrary is 
expressed. 



CHAP, v.] STRESS, STRAIN, AND RESILIENCE. 53 

49. The axes of a strain are the three directions in the 
body, at right angles to each other, which coincide with 
the directions of the axes of the ellipsoids. Lines drawn 
in the body in these three directions will remain at right 
angles to each other when the body is restored to its 
unstrained condition. 

A cube with its edges parallel to the axes will be 
altered by the strain into a rectangular parallelepiped. 
Any other cube will be changed into an oblique parallele- 
piped. 

When the axes have the same directions in space after 
sua before the strain, the strain is said to be unaccompanied- 
by rotation. When such parallelism does not exist, the 
strain is accompanied by rotation, namely, by the rotation 
which is necessary for bringing the axes from their initial 
to theii' final position. 

The numbers which specify a strain are mere ratios, 
and are therefore independent of units. 

50. When a body is under the action of forces which 
strain it, or tend to strain it; if we consider any plane 
section of the body, the portions of the body which it 
separates are pushing each other, pulling each other, or 
exerting some kind of force upon each other, across the 
section, and the mutual forces so exerted are equal and 
opposite. The specification of a stress must include a 
specification of these forces for all sections, and a body is 
said to be homogeneously stressed when these forces are 
the same in direction and intensity for all parallel sec- 
tions. We shall suppose stress to be homogeneous, in 
what follows, unless the contrary is expressed. 

51. When the force-action across a section consists of 



54 UNITS AND PHYSICAL CONSTANTS. [chap. 

a simple pull or push normal to the section, the direction 
of this simple pull or push (in other words, the normal to 
the section) is called an axis of the stress. A stress (like 
a strain) has always three axes, which are at right angles 
to one another. The mutual forces across a section not 
perpendicular to one of the three axes are in general 
partly normal and partly tangential — one side of the sec- 
tion is tending to slide past the other. 

The force per unit area which acts across any section is 
called the intensity of the stress on this section, or simply 
the stress on this section. The dimensions of " force per 

unit area," or — are - —,, which we shall therefore call 

area LT2' 

the dimensions 0/ stress. 

52. The relation between the stress acting upon a body 
and the straih produced depends upon the resilience of 
the body, which requires in general 21 numbers for its 
complete specification. When the body has exactly the 
same properties in all directions, 2 numbers are sufficient. 
These specifying numbers are usually called coefficients of 
elasticity; but the word elasticity is used in so many 
senses that we prefer to call them coefficients 0/ resilierice. 
A coefficient of resilience expresses the quotient of a 
stress (of a given kind) by the strain (of a given kind) 
which it produces. A highly resilient body is a body 
which has large coefficients of resilience. Steel is an 
example of a body with large, and cork of a body with 
small, coefficients of resilience. 

In all cases (for solid bodies) equal and opposite strains 
(supposed small) require for their production equal and 
opposite stresses. 



v.] STRESS, STRAIN, AND RESILIENCE. 55 

53. The coefficients of resilience most frequently re- 
ferred to are the three following : — 

(1) Resilience of volume, or resistence to hydrostatic 
compression. If V be the original and Y -v the strained 

volume, — is called the compression, and when the body 

is subjected to uniform normal pressure P per unit area 
over its whole surface, the quotient of P by the compres- 
sion is the resilience of volume. This is the only kind of 
resilience possessed by liquids and gases. 

(2) Young's modulus, or the longitudinal resilience of 
a body which is perfectly free to expand or contract 
laterally. In general, longitudinal extension produces 
lateral contraction, and longitudinal compression produces 
lateral extension. Let the unstrained length be L and 

the strained length Ij±1, then - is taken as the measure 

of the longitudinal extension or compression. The stress 
on a cross section (that is, on a section to which the stress 
is normal) is called the longitudinal stress, and Young's 
modulus is the quotient of the longitudinal stress by the 
longitudinal extension or compression. If a wire of cross 
section A sq. cm. is stretched with a force of F dynes, 
and its length is thus altered from L to L + /, the value 

■p T 

of Young's modulus for the wire is - . y. 

(3) " Simple rigidity " or resistance to shearing. This 
requires a more detailed explanation. 

54'. A -shear may be defined as a strain by which a 
sphere of radius unity is converted into an ellipsoid of 
semiaxes 1, 1+6, I -e; in other words, it consists of an 



56 



UNITS AND PHYSICAL CONSTANTS. [chap. 



extension in one direction combined with an eqnal com- 
pression in a perpendicular direction. 

55. A unit square (Fig. 1) whose diagonals coincide 
with these directions is altered by the strain into a 
rhombus whose diagonals are (1 + e) J2 and (1 - e) ,J% 
and whose area, being half the product of the diagonals, 
is 1-6^, or, to the first order of small quantities, is 1, 
the same as the area of the original square. The length 
of a side of the rhombus, being the square root of lihe 





Fig.i 



rt^.z 



sum of the squares of the semi-diagonals, is found to be 

Vl + e^ or 1 + ^\ and is therefore, to the first order of 
small quantities, equal to a side of the original square. 

56. To find the magnitude of the small angle which a 
side of the rhombus makes with the corresponding side of 
the square, we may proceed as follows :— Let ach (Fig. 2) 
be an enlarged representation of one of the small tri- 



angles in. Fig. 1 . Then we have ab = ^, cb = ^e J2 = 



Ji' 



TT 



angle cha = -. Hence the length of the perpendicular cd 



IT 



is ch sin^ = -. - — ^ = - : and since ad is ultimately 

equal to a6, we have, to the first order of small quan- 
tities. 



v.] 



STRESS, STRAIN, AND RESILIENCE. 



57 



, , cd ie 
angle cab = -3= ^ = e. 

ad ^ 



TT 



The semi-angles of the rhombns are therefore - ± e, 



IT 



and the angles of the rhombus are - ± 2e ; in other 

words, each angle of the square has been altered by the 
amount 2e, This qiuintity 2e is adopted as the measure 
of the shear, 

57. To find the perpendicular distance between oppo- 
site sides of the rhombus, we have to multiply a side by 
the cosine of 2e, which, to the first order of small quan- 
tities, is 1. Hence the perpendicular distance between 
opposite sides of the square is not altered by the shear, 
and the relative movement of these sides is represented 





^^ff ^ rig. 4 

by supposing one of them to remain fixed, while the 
other slides in the direction of its own length through a 
<iistance of 2e, as shown in Fig. 3 or Fig. 4. Fig. 3, in 
fact, represents a shear combined with right-handed rota- 
tion, and Fig. 4 a shear combined with left-handed rota- 
tion, as appears by comparing these figures with Fig. 1, 
which represents shear without rotation. 

58. The square and rhombus in these three figures may 
be regarded as sections of a prism whose edges are per- 
pendicular to the plane of the paper, and figures 3 and 4 



68 UNITS AND PHYSICAL CONSTANTS. [chap. 

show that (neglecting rotation) a shear consists in the 
relative sliding of parallel planes without change of dis- 
tance, the amount of this sliding being proportional to the 
distance, and being in fact equal to the product of the 
distance by the numerical measure of the shear. A good 
illustration of a shear is obtained by taking a book, and 
making its leaves slide one upon another. 

It may be well to remark, by way of caution, that the 
selection of the planes is not arbitrary as far as direction 
is concerned. The only planes which are affected in the 
manner here described are the two sets of planes which 
make angles of 45° with the axes of the shear (these axes 
being identical with the diagonals in Fig. 1). 

59. Having thus defined and explained the term 
" shear," which it will be observed denotes a particular 
species of strain, we now proceed to define a shearing 
stress. 

A shearing stress may be defined as the combination of 
two longitudinal stresses at right angles to each other, 
these stresses being opposite in sign and equal in magni- 
tude ; in other words, it consists of a pull in one direction 

combined with an equal thrust in a 
perpendicular direction. 

60. Let P denote the intensity 
of each of these longitudinal 
stresses; we shall proceed to cal- 
culate the stress upon a plane in- 
I ■*' clined at 45° to the planes of these 
. stresses. Consider a unit cube so 

" ' taken that the pull is perpendicular 

to two of its faces, AB and DC (Fig. 5), and the thrust 




v.] STRESS, STRAIN, AND RESILIENCE. 59 

is perpendicular to two other faces, AD, BC. The forces 
which hold the half-cube ABC in equilibrium are — 

(1) An outward force P, uniformly distributed over the 
face AB, and having for its resultant a single force P 
acting outward applied at the middle point of AB. 

(2) An inward force P, having for its resultant a single 
force P acting inwards at the middle point of BC. 

(3) A force applied to the face AC. 

To determine this third force, observe that the other 

two forces meet in a point, namely, the middle point of 

AC, that their components perpendicular to AC destroy 

one another, and that their components along AC, or 

p 
rather along CA, have each the magnitude -- - ; hence 

their resultant is a force P ^2, tending from C towards A. 
The force (3) must be equal and opposite to this. Hence 
each of the two half-cubes ABC, ADC exerts upon the 
other a force P ^2, which is tangential to their plane of 
separation. The stress upon the diagonal plane AC is 
therefore a purely tangential stress. To compute its 
intensity we must divide its amount P ^2 by the area of 
the plane, which is ^2, and we obtain the quotient P. 
Similar reasoning applies to the other diagonal plane BD. 
P is taken as the measure of the shearing stress. The above 
discussion shows that it may be defined as the intensity of 
tlie stress either on the planes of purely normal stress^ or 
on the planes of purely tangential stress, 

61. A shearing stress, if applied to a body which has 
the same properties in all directions (an isotrojnc body), 
produces a simple shear with the same axes as the stress ; 
for the extension in the direction of the pull will be equal 
to the compression in the direction of the thrust ; and in 



60 



UNITS AND PHYSICAL CONSTANTS. [chap. 



the third direction, which is perpendicular to both of 
these, there is neither extension nor contraction, since 
the transverse contraction dae to the pull is equal to the 
transverse extension due to the thrust. 

A shearing stress applied to a body which has not the 
same properties in all directions produces in general a 
shear with the same axes as the stress, combined with 
some other distortion. 

In both cases, the quotient of the shearing stress by the 
shear produced is called the resistance to shea/ring. In the 
case of an isotropic body, it is also called the simple rigidity, 

62. The following values of the resilience of liquids 
under compression are reduced from those given in 
Jamin, ^*Oours de Physique," 2nd edition, torn, i pp. 
168 and 169 :— 









Compression for 




Temp 


Coeflaclent of 


one Atxnosphere 




Cent. 


Resilience. 


(m^adyne per 
square centim.) 


Mercury, - 


00 


3-436x1011 


1 
2-91 X 10-« 


Water, 


0-0 


2 02 xlO^o 


4-96 X 10-» 


,. - - 


1-6 


1-97 „ 


508 „ 


.J - - 


41 


2-03 „ 


4-92 „ 


»» 


10-8 


2-11 „ 


4-73 ,, 


,, 


13-4 


2-13 „ 


4-70 „ 


, . 


18-0 


2-20 ,, 


4-65 „ 


)> 


25 


2-22 „ 


4-50 „ 


»» 


34-5 


2-24 „ 


4-47 „ 


j» " ■ 


43 


2-29 „ 


4*36 „ 


»j " ~ 


53-0 


2-30 „ 


4-35 „ 




r 0*61 


9-2 X 10» 


109x10-* 


Ether, - 


- 00 V 


7-8 „ 


1 -29 „ 




U4-0j 


7-2 „ 


1-38 „ 


Alcohol, 


7-3 { 
\ 13-1 i 


1-22 xlO^o 
112 „ 


8-17 „ 
8-91 X 10-» 


Sea Water, - 


17-5 


2-33 „ 


4-30 „ 




^.] 



STRESS, STRAIN, AND RESILIENCE. 



61 



63. The following are reduced from the results ob- 
tained by Amaury and Descamps, " Comptes Bendus/' 
torn. Ixviii. p. 1564 (1869), and are probably more 
accurate than the foregoing, especially in the case of 
mercury : — 







Coefficient of 
Resilience. 


Compression for 

one megadyne per 

square centim. 


Distilled Water, 

Alcohol, - 
/ „ ... 

Ether, 

»> - - - 
/ Bisulphide of Carbon, 

Mercury, - 


o 

15 

15 

14 
14 
15 


2-22 X 1010 
1-21 „ 

111 „ 
9-30x109 
7-92 „ 
1-60x1010 
5-42 X 1011 


4-51 X 10-5 
8-24 „ 
8-99 „ 
1-08x10-4 
1*26 „ 
6-26x10-5 
1-84x10-6 



64. The following values of the coefficients of resilience 
'or solids are reduced from those given in my own papers 
o the Royal Society (see « Phil. Trans.," Dec. 5th, 1867, 
^ 369), by employing the value of g at the place of ob- 
^rvation^ namely, 981*4. 





Young's 
Modulus. 


Simple 
Rigi(fity. 


Resilience of 
Volume. 


Density 


CHass, flmt. 


6-03 xlOii 


2-40 X 1011 


4-15 xlOii 


2-942 


-Another specimen 


5-74 „ 


2-35 „ 


3-47 „ 


2-935 


^rass, drawn, - 


1-075x101^ 


3-66 „ 




8-471 


«teel,- 


2-139 „ 


8-19 ,, 


1-841x101^ 


7-849 


Xx)n, wrought, - 


1-963 „ 


7-69 „ 


1-456 „ 


7-677 


„ cast,- 


1-349 „ 


5-32 „ 


9-64 xlOii 


7-236 


Copper, 


1-234 „ 


4-47 „ 


1-684x101*^ 


8-843 



65. The resilience of volume was not directly observed, 
^ut was calculated from the values of " Young's modulus " 
^ud "simple rigidity,'* by a formula which is strictly true 



62 



UNITS AND PHYSICAL CONSTANTS. [chap. 



for bodies which have the same properties in all direc- 
tions. The contraction of diameter in lateral directions 
for a body which is stretched by purely longitudinal stress 
was also calculated by a formula to which the same 
remark applies. The ratio of this lateral contraction to 
the longitudinal extension is called " Poisson's ratio," and 
the following were its vs^lues as thus calculated for the six 
bodies experimented on : — 



Glass, flint, '258 

Another specimen, '229 

Brass, drawn, '469 (?) 

Steel, -310 



Iron, wrought, '275 

,, cast, '267 

Copper, -378 



Kirchhoff has found for steel the value '294, and Clerk 
Maxwell has found for iron '267. Comu ("Oomptes 
Rendus," August 2, 1869) has found for different speci- 
mens of glass the values -225, '226, '224, -257, '236, -243, 
•250, giving a mean of '237, and maintains (with many 
other continental savants) that for all isotropic solids 
(that is, solids having the same properties in all direc- 
tions) the true value is ^. 

66. The following values of Poisson's ratio have been 
found by Mr. A. Mallock (" Proc. Roy. Soc," June 19, 
1879) :— 



Steel, 

Brass, 

Copper, 

Lead, 

Zinc (rolled), 

,, (cast),... 

Ebonite, 



•253 
•325 
•348 
•375 
•180 
•230 
•389 



Ivory, '50 

India Rubber, '50 

Paraffin, '50 

Plaster of Paris,. . . . '181 

Cardboard, '2 

Cork, -00 



Boxwood,... 
Beechwood, , 
White Pine, 



Radial 

due to 

LongitudinaL 

•42 

•53 

•486 



Longitudinal 
due to 
Radial. 

•406 

•408 

•372 



In 

Cross 

Section. 



•227 



v.] STRESS, STRAIN, AND RESILIENCE. 63 

The heading "Radial due to Longitudinal" means 
that the applied force is longitudinal (that is, parallel to 
the length of the tree) and that the contraction along a 
radius of the tree is compared with the longitudinal 
extension. 

67. The following are reduced from Sir W. Thomson's 
results ("Proc. Roy. Soc," May, 1865), the value of g 
being 981-4:— 

Simple Rigidity. 

Brass, three specimens, 4*03 3*48 3*44) j^u 

Copper, two specimens, 4*40 4*40 { 

Other specimens of copper in abnormal states gave 
i-esults ranging from 3*86 x 10^^ to 4*64 x 10^^ 

The following are reduced from Wertheim's results 
(" Ann. de Ohim.," ser. 3, tom. xxiii.), g being taken as 
981:— 

Different Specimens of Glass {Crystal), 

Young's Modulus, 3*41 to 4*34, mean 3*96 ^ 

Simple Rigidity, 1 -26 to 1 '66 „ 1 '48 }- x 10" 

Volume Resilience 3*50 to 4*39 ,, 3 '89 J 



Different Specimens of Brass. 

] 



Young's Modulus, .... 9 '48 to 10 44, mean 9 '86 

Simple Rigidity, 3*53 to 3*90 „ 3*67 }- x lO^i 

Volume Resilience,.. 1002 to 10-85 „ 10-43 



68. Savart's experiments on the torsion of brass wire 
(« Ann. de Chim.," 1829) lead to the value 3-61 x 10" for 
simple rigidity. 

Kupffer's values of Young's modulus for nine different 
specimens of brass range from 7*96 x 10" to 11*4 x 10^^, 
the value generally increasing with the density. 

For a specimen, of density 8*4465, the value was 
10-58 X 10". 



64 UNITS AND PHYSICAL CONSTANTS, [chap. v. 

For a specimen, of density 8*4930, the value was 
11-2x10". 

The values of Young's modulus found by the same experi- 
menter for steel, range from 20*2 x 10" to 21*4 x 10". 

69. The following are reduced from Kankine's '' Rules 
and Tables," pp. 195 and 196, the mean value being 
adopted where different values are given : — 

Tenacity. Young's Modulus. 

Steel Bars, 7'93xlO» 2*45 x lO^^ 

IronWire 6'86 „ 1745 „ 

Copper Wire, 4*14 „ 1-172 „ 

BrassWire, 338 „ 9-81 x lO" 

Lead, Sheet, 2-28xl0« 5*0 x IQio 

Tin, Cast, 3*17 „ 

Zinc, 5-17 ,, 

Ash, 1-I72xl0» 1-10 xlQii 

Spruce, 8-55 x lO^ I'lO 

Oak, l-026xl0» 1-02 



99 

»» 



Glass, 6-48xl0« 6*52 x IQii 

Brick and Cement,.. 2*0 x 10" 

The tenacity of a substance may be defined as the 
greatest longitudinal stress that it can bear without tear- 
ing asunder. The quotient of the tenacity by Young's 
modulus will therefore be the greatest longitudinal exten- 
sion that the substance can bear. 



65 



CHAPTER VI. 

ASTRONOMY. 

Size and Figure of the Earth. 

70. According to the latest determination, as pub- 
liahed by Capt. Clarke in the " Philosophical Magazine " 
£or August, 1878, the semiaxes of the ellipsoid which 
most nearly agrees with the actual earth are, in feet, 

a = 20926629, b - 20925105, c = 20854477, 

-which, reduced to centimetres, are 

a = 6-37839 x lO^, b = 6-37792 x lO^, c = 6-35639 x lO^, 

giving a mean radius of 6*3709 x 10^, and a volume of 
1-0832 x 1027 cubic centims. 

The ellipticities of the two principal meridians are 

1 and 1 



289-5 296-8 

The longitude of the greatest axis is S"" 15' W The mean 
ieiigth of a quadrant of the meridian is 1*00074 x 10^. 

The length of a minute of latitude is approximately 
1 85200 - 940 cos. 2 lat. of middle of arc. 

The mass of the earth, assuming Baily's value 5*67 for 
mean density, is 6*14 x 10^^ grammes. 

£ 



r, r 



©"-' 



I 



66 UNITS AND PHYSICAL CONSTANTS. [chap. 

Day and Year. 

Sidereal day, 86164 mean solar seconds. 

Sidereal year, 31,558,150 „ 

Tropical year, 31,556,929 „ 

2x 1 

Angular velocity of earth's rotation, = ^^5=^0. 

861o4 13713 

Velocity of earth in orbit, about 2960600 „ 

*^*Si iXtiot T.'*!r '"!} 3-3908 dynes per g«nune. 

Attraction in Astronomy, 

71. The mass of the moon is the product of the earth's 
mass by -011364, and is therefore to be taken as 
6-98 X 102^ grammes, the doubtful element being tiie 
earth's mean density, which we take as 5*67. 

The mean distance of the centres of gravity of the 
earth and moon is 60*2734 equatorial radii of the earth 
—that is, 3-8439 x lO^^ centims. 

The mean distance of the sun from the earth is about 
1-487 X 10^^ centims., or 92-39 million miles, correspond- 
ing to a parallax of 8"-848.* 

The intensity of centrifugal force due to the earth'i 

"T ) 

noting the mean distance, and T the length of the sidereaEI 
year, expressed in seconds. This is equal to the accelera- 
tion due to the sun's attraction at this distance. Puttin 
for r and T their values, 1-487 x lO^s and 3-1558 x 10 

we have 1 -r^ 1 r = -5894. 




* This value of the mean solar parallax was determined by Prt^ -^ 
feesor Newcomb, and was adopted in the '* Nautical Almanac 
lor 1882. (See Art. 86 for a later determination.) 



VI.] ASTRONOMY. 67 

This is about , ^^^ of the value of g at the earth's 

1660 ^ 

surface. 

The intensity of the earth's attraction at the mean dis- 
tance of the moon is about 

-^?1- or -2701. 
(60-27)2 

This is less than the intensity of the sun's attraction upon 
the earth and moon, which is *5894 as just found. Hence 
the moon's path is always concave towards the sun. 

72. The mutual attractive force F between two masses 
m and m', at distance I, is 

where is a constant. To determine its value, consider 
the case of a gramme at the earth's surface, attracted by 
the earth. Then we have 

F = 981, m = l, m' = 6-14 x 1027, ; = 6-37 x lO^; 
whence we find 

C = ^= -^ 

108 1.543x107* 

TTo find the mass m which, at the distance of 1 centim. 

^rom an equal mass, would attract it with a force of 1 

<iyne, we have 1 = Qrn^ ; 

'vrhence m = . - = 3928 grammes. 

73. To find the acceleration a produced at the distance 
of I centims. by the attraction of a mass of m grammes, 

^e have a = — - = C— , 

m l^ 

where C has the value 6*48 x 10"® as above. 



68 UNITS AND PHYSICAL CONSTANTS. [chap. 

To find the dimensions of C we have C = — , where the 

m 

dimensions of a are LT~^. 

The dimensions of C are therefore 

1^2 3^-1 LT-2 . i-hat is, L» M-^ T'l 

74. The equation a = C^ shows that when a = 1 and 

^=1, 971 must equal — ; that is to say, the mass which 

produces unit acceleration at the distance of 1 centimetre 
is 1*543 X 10^ grammes. If this were taken as the unit 
of mass, the centimetre and second being retained as the 
units of length and time, the acceleration produced by the 
attraction of any mass at any distance would be simply 
the quotient of the mass by the square of the distance. 

It is thus theoretically possible to base a general system 
of units upon two fundamental units alone ; one of the 
three fundamental units which we have hitherto employed 
being eliminated by means of the equation 

mass = acceleration x (distance)^, 
which gives for the dimensions of M the expression 
U T-l 

Such a system would be eminently convenient in astro- 
nomy, but could not be applied with accuracy to ordinary 
terrestrial purposes, because we can only roughly compare 
the earth's mass with the masses which we weigh in our 
balances. 

75. The mass of the earth on this system is the 

product of the acceleration due to gravity at the earth's 

surface, and the square of the earth's radius. This 

product is 

981 X (6-37 X 108)2 = 3-98 X IO20, 



\ 



VI.] ASTRONOMY. 69 

and is independent of determinations of the earth's 
density. 

The new unit of force will be the force which, acting 
iipon the new unit of mass, produces unit acceleration. 
It will therefore be equal to 1*543 x 10^ dynes; and its 
dimensions will be 

mass X acceleration = (acceleration)^ x (distance)^ 

= L* T-*. 

76. If we adopt a new unit of length equal to I 
oentims., and a new unit of time equal to t seconds, while 
we define the unit mass as that which produces unit 
acceleration at unit distance, the unit mass will be 

Pt'^ X 1*543 X 10^ grammes. 

If we make I the wave-length of the line F in vacuo, 
say, 4-86 x 10"*, 

and t the period of vibration of the same ray, so that 

is the velocity of light in vacuo, say, 

3 X 1010, 

/ 7 

the value of P t"^ or I 



i^'- 



4-374 X 10i«, 
and the unit mass will be the product of this quantity 
into 1*543 x 10^ grammea This product is 6*75 x 10^3 
grammes. 

The mass of the earth in terms of this unit is 
3*98 x 1020 ^ (4*374 x 10i«) = 9100, 
and is independent of determinations of the earth's 
density. 



CHAPTER VII. 

VELOCITY OF SOUND. 

77. The propagation of sound through any medium is 
due to the elasticity or resilience of the medium ; and 
the general formula for the velocity of propagation s is 



8 






where D denotes the density of the medium, and E the 
coefficient of resilience. 

78. For air, or any gas, we are to understand by E the 

quotient 

increment of pressure 
corresponding compression ' 
that is to say, if P, P + p be the initial and final pres- 
sures, and V, V - V the initial and final volumes, p and v 
being small in comparison with P and V, we have 

V V 

V 

If the compression took place at constant temperature, 

we should have 

| = |.andE = P. 

But in the propagation of sound, the compression is 
efiected so rapidly that there is not time for any sensible 
part of the heat of compression to escape, and we have 



CHAP. VII.] VELOCITY OF SOUND. 71 

where y = 1'41 for dry air, oxygen, nitrogen, or hydrogen. 
The value of — for dry air at f Cent, (see p. 46) is 

(1 + -003660 ^ 7-838 + lO^. 
Hence the velocity of sound through dry air is 

«= W >/l-41x(l + '00366<)x 7-838 

= 33240 VI + .00366«; 
or approximately, for atmospheric temperatures, 

8 = 33240 + 60<. 

79. In the case of any liquid, E denotes the resilience 
of volume.* 

For water at 8***1 C. (the temperature of the Lake of 
G^eneva in Colladon's experiment) we have 
E = 2-08 X 1010, D = 1 sensibly ; 



V 



1= VE = 144000, 



the velocity as determined by CoUadon was 143600. 

80. For the propagation of sound along a solid, in the 
form of a thin rod, wire, or pipe, which is free to expand 
or contract laterally, E must be taken as denoting Young's 
modulus of elasticity.* The values of E and D will be 
different for different specimens of the same material. 
Employing the values given in the Table (§ 64), we have 

* Strictly speaking, E should be taken as denoting the resili- 
ence for sudden applications of stress — so sudden that there is 
not tim^e for changes of temperature produced by the stress to be 
sensibly diminished by conduction. This remark applies to both 
§§ 79 and 80. For the amount of these changes of temperature, 
see a later section under Heat. 



72 



UNITS AND PHYSICAL CONSTANTS. [chap. 





Values of E. 


Values of 
D. 


Values of ^5, 
or velocity. 


Glass, first specimen, 
„ secona specimen, 

Brass, 

Steel, 

Iron, wrought, - 
„ cast, - 

Copper, , - 


6-03 xlO" 

6-74 

1-075x1112 

2-139 „ 
1-963 „ 
1-349 „ 
1-234 „ 


2-942 
2-935 
8-471 
7-849 
7-677 
7-235 
8-843 


4-63 X W 
4-42 „ 
3-66 „ 
5-22 „ 
6-06 „ 
4-32 „ 
3-74 „ 



81. If the density of a specimen of red pine be '5, and 
its modulus of longitudinal elasticity be 1-6 x 10^ pounds 
per square inch at a place where g is 981, compute the 
velocity of sound in the longitudinal direction. 

By the table of stress, page 4, a pound per square inch 
{g being 981) is 6-9 x 10* dynes per square centim. Hence 
we have for the required velocity 

/E^ / l-6x 100x6-9x1 0* 

centims. per second. 

82. The following numbers, multiplied by 10^, are the 
velocities of sound through the principal metals, as 
determined by Wertheim : — 



= 4-7 X 10^ 





At 20' C. 


At 100" C. 


At 200* C. 


Lead, 


1-23 


1-20 


• • • 


Gold, 


1-74 


1-72 


1-73 


Silver, 


2-61 


2-64 


2-48 


Copper, - 


3-56 


3-29 


2-95 


Platmum, 


2-69 


2-57 


2-46 


Iron, 


6-13 


5-30 


4-72 


Iron Wire (ordinary), 


4-92 


510 


• • • 


Cast Steel, 


4-99 


4-92 


4-79 


Steel Wire (English), 


4-71 


5-24 


5-00 


» " " 


4-88 


5 01 


• • • 



vn.] 



VELOCITY OF SOUND. 



73 



The following velocities in wood are from the observa- 
tions of Wertheim and Chevandier, " Comptes Rendus," 
1846, pp. 667 and 668 :— 





Along Fibres. 


Radial 
Direction. 


Tangential 
Direction. 


Pine, 


3*32 xO« 


2-83 X W 


l-69xl0« 


Beech, - 


3*34 ,, 


3-67 „ 


2-83 „ 


Witch-Ehn, - 


3-92 „ 


3-41 „ 


2-39 „ 


Birch, - 


4-42 „ 


2-14 „ 


303 „ 


Fir, - - - 


4-64,, 


2-67 „ 


1-57 „ 


Acacia, - 


4-71 „ 






AspeD, - 


6-08 „ 







Musical Strings. 

83. Let M denote the mass of a string per unit length, 

F „ stretching force, 

L „ length of the vibrating portion ; 

then the velocity with which pulses travel along the 
string is 



"=Vm' 



and the number of vibrations made per second is 



V 



Bxample. 

For the four strings of a violin the values of M in 
;grammes per centimetre of length are 

•00416, -00669, -0106, -0266. 
T!he values of n are 

660, 440, 293J, 195|; 



74 UNITS AND PHYSICAL CONSTANTS. [oHAP.vn, 

and the common value of L is 33 centims. Hence the 
values of v or 2Ln are 

43560, 29040, 19360, 12910 
centims. per second ; and the values of F or Mt;^, in 
dynes, are 

7-89 X 10«, 5-64 X 10«, 3-97 x 10«, 4-43 x 10». 

Faintest Audible Sound. 

84. Lord Rayleigh (" Proc. R. S.," 1877, vol. xxvi p. 
248), from observing the greatest distance at which a 
whistle giving about 2730 vibrations per second, and blown 
by water-power, was audible without effort in the middle 
of a fine still winter's day, calculates that the maximum 
velocity of the vibrating particles of air at this distance 
from the source was *0014 centims. per second, and that 
the amplitude was 8*1 x 10"^ centims., the calculation 
being made on the supposition that the sound spreads- 
uniformly in hemispherical waves, and no deduction being 
made for dissipation, nor for waste energy in blowing. 



75 



CHAPTER VIII. 

LIGHT. 

85. All kinds of light are believed to have the same 
velocity in vacuo. The velocity of light of given re- 

frangibility in any medium is - of .its velocity in vacuo, 

A* 
/x denoting the absolute index of refraction of that medium 

for light of the given refrangibility. 

Light of given refrangibility is light of given wave- 
frequency. Its wave-length in any medium is the 
quotient of its velocity in that medium by its wave- 
frequency. If n denote the wave-frequency (that is to 
say, the number of waves which traverse a given point 

in a second), the wave-length in any medium will be — 

of the velocity in vacuo. 

The absolute index of refraction for ordinary air is 
about 1*00029. More accurate statements of its value 
will be found in Arts. 94-96. 

86. The best determination of the velocity of light is 
that made by Professor Newcomb at Washington in 1882 
(" Astron. Papers of Amer. Ephem.," vol. ii. parts iii 
and iv. 1885). The method employed was that of the 
revolving mirror, the distance between the revolving and 



76 



UNITS AND PHYSICAL CONSTANTS. [chap. 



the fixed mirror being ia one portion of the observations 
2550 metres, and in the remaining portion 3720 metres. 
The resulting velocity in vacuo is 

2-99860 X 10^0 centims. per sea 
The following summary of results is from Professor 
Newcomb's paper, page 202 : — 



km. per. sec 
299910 
299853 



I 



299860 

299810 
298000 
298500 
300400 
299990 
301382 



Michelson, at Naval Academy, in 1879, 

Michelson, at Cleveland, 1882, 

Newcomb, at Washington, 1882, using only 
results supposed to be nearly free from 
constant errors, 

Newcomb, including all determinations, 

Foucault, at Paris, in 1862, 

Comu, at Paris, in 1874, 

Comu, at Paris, in 1878, 

This last result as discussed by Listing, 

Young and Forbes, 1880-81, 

Professor Newcomb remarks (page 203) that the value 
299860 km. per sec. for the velocity of light, combined 
with Clark's value 6378 '2 km. for the earth's equatorial 
radius, and Nyren's value 20"'492 for the constant of 
aberration, gives for the solar pai*allax the value 8" '7 94. 

87. The following are the wave-lengths adopted by 
Angstrom for the principal Fraunhofer lines in air at 760 
millims. pressure (at XJpsal) and 16* C. : — 

Centims. 

A 7-604 xlO-5 

B 6-867 

C 6-56201 

Mean of Hues D .5-89212 

E 5-26913 

F 4-86072 

G 4-30725 

Hi 3-96801 

Hs 3-93300 



vm.] LIGHT. 77 

These nambers will be approximately converted into 
the corresponding wave-lengths in vacuo by multiplying 
them by 1-00029. 

88. Assuming 3 x 10^^ to be the velocity of light in 
air, and neglecting the difference of velocity between 
the more and less refrangible rays, we obtain the follow- 
ing frequencies by dividing the common velocity by 
Angstrom's values of the wave-lengths : — 

Vibrations per Second. 

A 3045x101* 

B 4-369 „ 

C 4-672 „ 

D 5-092 „ 

E 5-693 „ 

F 6-172 „ 

G 6-965 „ 

Hi 7-560 „ 

H, 7-628 „ 

According to Langley (" Com. Hen./' Jan., 1886), the 
solar spectrum extends beyond the red as far as wave- 
length 27 X 10"^ and the radiation from terrestrial bodies 
at temperatures below 100° extends as far as wave-length 
150 X 10"^ The frequencies corresponding to these two 
wave-lengths are 1*1 x 10^* and 2 x 10^^. 

INDICES OF REFRACTION OF SOLIDS. 

89. Dr. Hopkinson ("Proc. R. S.," June 14, 1877) has 
determined the indices of refraction of the principal 
varieties of optical glass made by Messrs. Chance, for the 
fixed lines A, B, C, D, E, 6, F, (G), G, h, H^. By D is 
to be understood the more refrangible of the pair of 
sodium lines ; by b the most refrangible of the group of 
magnesium lines ; by (G) the hydrogen line near G. 



78 UNITS AND PHYSICAL CONSTANTS. [chap. 

In connection with the results of observation, he 
employs the empirical formula 

fjL-l =a{l -hbx^l+cx)}, 
where a; is a numerical name for the definite ray of which 
/A is the refractive index. In assigning the valae of x, 
four glasses— hard crown, soft crown, light flint, and 
dense flint — were selected on account of the good accord 
of their results ; and the mean of their indices for any 
given ray being denoted by /I, the value assigned to x for 
this ray is /a - fip where /Ip denotes the value of /x for the 
line F. 

The value of /a as a function of A., the wave-length in 
10"* centimetres, was found to be approximately 

fx = 1-538414 + 0-0067669^2 - 0-00017341 

A A 

+ 0-000023-1. 

A^ 
The following were the results obtained for the diflerent 
specimens of glass examined : — 

Hard Crown, 1st specimen, density 2*48575. 

a=0-523145, 5 = 1-3077, c= -2-33. 

Means of observed values of fi. 



A 1-611755 
E 1-520324 
G 1-528348 



B 1-513624 
b 1-520962 
h 1-530904 



C 1-514571; D 1-517116; 
F 1-523145; (G) 1*527996; 
Hi 1-532789. 



Soft Crown, density 2-55035. 

a =0-5209904, 6=1*4034, c=-l-58. 

Means of observed values of fi. 



A 1-508956 
E 1*518017 
G 1*626692 



B 1-510918 
h 1-518678 
h 1-529360 



C 1-511910; D 1*514580; 
F 1*620994; (G) 1*626208; 
Hj 1*531415. 



vin.] 



LIGHT. 



79 



Extra Light Flint Glass, density 2*86636. 

a=0-549123, 6=1-7064, c=-0198. 

Means of observed values of fi» 



A 1-634067 

D 1-541022 

P 1-549125 

h 1*559992 



B 1 -536450 

E 1*545295 

(G) 1-555870 

Hi 1-562760. 



C 1*537682 

b 1*546169 

G 1*556375 



LigM Flint Glass, density 3*20609. 

a=0*583887, 6 = 1*9605, c=0-53. 

Means of observed values of fi. 



B 1*568558 
K 1-579227 
<G) 1*592184 
H, 1-600717. 



C 1*570007 

6 1*580273 

G 1*592825 



D 1*574013 
F 1*583881 
k 1*597332 



Dense Flint, density 3*65865. 

a = 0*634744, 6 = 2-2694, c = l*48. 

Means of observed values of /<. 



B 1*615704 

£ 1-628882 

(G) 1*645268 

Hi 1*656229. 



C 1*617477 

6 1*630208 

G 1*646071 



D 1*622411 
F 1*634748 
h 1*651830 



Extra Dense Flint, density 3 '88947. 

a=0*664226, 6=2*4446, c=l*87. 

Means of observed values of fi. 



A 1-639143 
D 1-650374 
F 1-664246 
A 1-683575 



B 1*642894 
E 1-657631 
(G)* 1*676090 
Hi 1*688590. 



1*644871 
6 1*659108 
G 1*677020 



80 



UNITS AND PHYSICAL CONSTANTS. [chap. 



Double Extra Dense FlnU, density 4*42162. 

a=0-727237, ^=2*7690, c=2-70. 

Means of observed values of /jl. 



A 1-696531; 
D 1-710224; 
F 1-727257; 
h 1-751485. 



B 1-701080 

E 1-719081 

(G) 1-742058 



C 1-703485 

b 1-720908 

G 1-743210 



90. The following indices of rock salt, sylvin, and alam 
for the chief Fraunhofer lines are from Stefan's observer 
tions : — 



A 
B 

D 
E 
F 
G 
H 



Bock Salt 
at 17' 0. 

1-53663 

•53918 

•54050 

•54418 

-54901 

•55324 

-56129 

-56823 



Sylvin 
at 20* C. 

1-48377 

•48597 

•48713 

•49031 

•49455 

•49830 

•50542 

•51061 



Alum 
at 2V C. 

1-46057 

•45262 

-45359 

•45601 

•45892 

•46140 

•46563 

•46907 



91. Indices of other singly refracting solids — 

Index of Kind of 

Refraction. Light. 

Diamond, 2-470 D 

Fluor-spar, 1*4339 D 

Amber, 1*532 D 

Rosin, 1545 Red 

Copal, 1*528 Red 

Gum Arabic, 1*480 Red 

Peru Balsam 1 '593 D 

Canada Balsam,. 1-528 Red 



Observer. 

Schrauf. 
Stefan. 
Kohlrausch. 
Jamin. 



it 



>i 



Baden Powell. 
Wollaston. 



Hffhct of Temperature, 

According to Stefan, the index of refraction of glass 
increases by about -000002 for each degree Cent, of 



Tm.] LIGHT. 81 

increase of temperature, and the index of rock salt 
diminiahea hj aboat *000 037 for each degree of increase 
of temperature. 

92. Doubly refracting crystals : — 

Uniaacal Crystals, 

rw.»n.«« Bxtraordi- Kind 
"JJ^*J7 nary of Temp. Obaerver. 

luaex. Index. Light. 

Ice, 1*3060 1-3073 Red Reusch. 

Iceland-spar 1*65844 1*48639 D 24** v. d. WiUigen. 

Nitrate of Soda, 1*5854 1*3369 D 23** F. Kohlransoh. 

Quartz, 1-64419 1*55329 D 24** v. d. Willigen. 

Tourmaline, 1*6479 1*6262 Green 22° Heusser. 

Zircon, 1*92 1*97 Red de Senarmont. 

Biaocal Crystals, 

THBEB PRINCIPAL INDICES OF BEFRACTION FOB SODIUM LIGHT. 

Least. Intermediate. Greatest. Temp. Observer. 

Arragonite, 1*53013 1*68157 1*68589 Rudberg. 

Borax, 1*4463 1*4682 1*4712 23° Kohlrausch. 

Mica, 1*5609 1*5941 1*5997 23° 

Nitre, 1*3346 15056 15064 16° Schrauf. 

Selenite, 1*52082 1*52287 1*53048 17° v. Lang. 

^5?fs^tic)}l-9S«5 2*0383 2*2405 16° Schrauf. 
Topaz, 1*61161 1*61375 1*62109 Rudberg. 

INDICES OF BEFRACTION FOR LIQUIDS. 

93. The following values of indices of refraction for 

liquids are condensed from Fraunhofer's determinations, 

as given by Sir John Herschel (" Enc. Met. Art.," Light, 

p. 415) :— 

fTafer, density 1*000. 

B 1*3309; C 1*3317; D 1'3336; 

E 1 3358 ; F 1 3378 ; G 1-3413 ; 

H 1*3442. 

F 



82 UNITS AND PHYSICAL CONSTANTS. [chap. 

OU of Turpmtine, density 0*8a5. 

B 1-4705; C 1-4715; D 1-4744; E 1-4784; 
F 1-4817; G 1 •488-2; H 1-4939. 

The following determinations of the refractive indices 
of liquids are from Gladstone and Dale's results, as given 
in Watf 8 " Dictionary of Chemistry," iii pp. 629-631 :— 

Sulphide of Carbon^ at temperature 11', 

A 1-6142; B 1-6207; C 16240; D 1-6333; 
E 1-6465; F 1-6584; G 1*6836; H 1-7090. 

Benzene, at temperature 10*5^ 

A 1-4879; B 1-4913; C 1-4931; D 1-4976; 
E 1-5036; F 15089; G 15202; H 1-5305. 

Chlorofomij at temperature 10**. 

A 1-4438; B 14457; C 1-4466; D 1-4490; 
E 1-4526; F 1-4555; G 14614; H 1-4661. 

Alcohol^ at temperature 15°. 

A 1-3600; B 13612; C 13621; D 13638; 
E 1-3661; F 1.3683; G 1-3720; H 1-3751. 

EiMr, at temperature 15°. 

A 1-3529; B 13545; C 1-3554; D 1-3666; 
E 1-3590; F 1-3606; G 13646; H 1-3683. 

Watery at temperature 15**. 

A 1-3284; B 1-3300; C 13307; D 13324; 
E 1-3347; F 1-3366; G 1*3402; H 1*3431. 



INDICES FOR OASES. 



94. Indices of refraction of air at 0** C. and 760 mm. 
for the principal fraunhofer lines. 



vra.] LIGHT. 83 





Acoordliig to Kettler. 


According to Lorenz. 


A 


1*00029286 


1*00028935 


B 


29350 


28993 


C 


29383 


29024 


D 


29470 


29108 


E 


29584 


29217 


F 


29685 


29312 


G 


29873 


29486 


H 


30026 


29631 



95. The formula established by the experiments of 
Biot and Arago for the index of i*efraction of air at 
various pressures and temperatures was 

_ . ^ -0002943 Ji 
^ Y+at * 760' 

a denoting the coefficient of expansion '00366, and h the 
pressure in millims. of mercury at zero. As the pressure 
of 760 millims. of such mercury at Paris is 1*0136 x 10^ 
dynes per sq. chl, the general formula applicable to all 
localities alike will be 

,_ -0002943 P 

^ l + *00366< • 10136 xlO«' 

where P denotes the pressure in dynes per sq. cm. This 
can be reduced to the form 

•0002903 P 



/*-! 



1 + oosae^ ' W 



96. According to Mascart, /a - 1 for any gas is pro- 
portional not to = but to 

h + ph^- 

where fi and a' are coefficients which vary from one gas 
to another. In the following table, the column headed fl^, 



84 UNITS AND PHYSICAL CONSTANTS. [chap. 

contains the indices for 0"* and 760 mm. at Paris. The 
next column contains the value of P multiplied bj 10^ (it 
being understood that h is expressed in millimetres), and 
the next column the value of a'. All these data are for 
the light of a sodium flame :— 

Ih /5xW a' 

Air, 1'0002927 7*2 -00382 

Nitrogen, 2977 8*5 382 

Oxygen, 2706 IM 

Hydrogen, 1387 -8-6 378 

Nitrous Oxide, 5159 88 388 

Nitrous Gas, 2975 7 367 

Carbonic Oxide, 3350 8*9 367 

Carbonic Acid, 4544 72 406 

Sulphurous Acid, .... 7036 25 460 

Cyanogen, 8216 277 

More recent, and probably more accurate observations, 

which will be published in vol. v. of " Travaux et 

Memoires du Bureau International des Poids et Mesures/' 

have been conducted by Benoit with Fizeau's dilatometer. 

They give 

1-0002923 

as the index of refraction of air for the D line at 0" 0. 

and 760 mm.; and for the temperature coefficient they 

give 

•003667, 

which is identical with the coefficient of expansion of air. 

The larger value, •00382, obtained by Mascart, is traced 

to imperfect measurement of temperature. 

Coefficient of Dispersive Power, 
97. Assuming Cauchy's formula 



/.-l=a(l + ^) 



vm.] 



LIGHT. 



85 



(where A is the wave-length), which is known to be 
approximately true for air within the limits of the visible 
spectrum, the constant b may be called the coefficient of 
dispersive power. Employing as the unit of length for 
A the 10""* of a centimetre, Mascart ("Ann, de V l^le 
Normale/' 1877, p. 62) has obtained the following values 
for b: — 

Coefficient of Dispersion. 

Air, -0058 

Nitrogen, -0067 

Oxygen, "0064 

Hydrogen, 0043 

Carbonic Oxide, '0075 

Carbonic Acid, 0052 

Nitrous Oxide, 0125 

Cyanogen, 0100 

According to Mascart, the ratio of dispersion to devia- 
tion for the two lines B and H is *024 for air, *032 for 
the ordinary ray in quartz, '038 for light crown glass, 
*040 for water, and *046 for the ordinary ray in Iceland- 
spar. 

Motation of Plane of Poh/rization, 

98. The rotation produced by 1 millim. of thickness of 
quartz cut perpendicular to the axis has the following 
values for different portions of the spectrum, according to 
the observations of Soret and Sarasin (" Com. Ben. 95," 
p. 635, 1882), the temperature of the quartz being 
20* C. :— 



Rotation. 

A 12'-668 

B 15^-746 

C 17**-318 

D, 2r-684 

Di 2r-727 



Rotation. 

E 27"-543 

F 32**-773 

G 42"-604 

H 6ri93 



86 UNITS AND PHYSICAL CONSTANTS. [cHAP.vni. 

According to the same observers, the rotation at 
f C. is equal to the rotation at 0*" C. multiplied by 
1+-000179<. 

Units of Illuminating Power. 

99. The British '^ Candle" is a spermaceti candle, 
{ inch in diameter (6 to the lb.), burning 120 grains 
per hour. 

The French " Carcel " is a lamp of specified construc- 
tion, burning 42 grammes of pure Colza oil per hour. 
One " carcel " is equal to about 9 J " candles." 

The unit adopted by the International Congress at 
Paris, April 1884, is a square centimetre of molten 
platinum at the temperature of solidification. The surface 
illuminated by it in photometric tests is to be normally 
opposite to the surface of the molten platinum. Accord- 
ing to the experiments of M. Yiolle the author of this 
unit, it is equal to 2*08 carcels. It is therefore about 
19| candles. 



87 



CHAPTER IX. 

HEAT. 

100. The unit of lieat is usually defined as the quantity 
of heat required to raise, by one degree, the temperature 
of unit mass of water, initially at a certain standard temr 
perature. The standard temperature usually employed is 
0** 0. ; but this is liable to the objection that ice may be 
present in water at this temperature. Hence 4** C. has 
been proposed as the standard temperature ; and another 
proposition is to employ as the unit of heat one hundredth 
part of the heat required to raise the unit mass of water 
from 0** to 100' C. 

101. According to Kegnault (" M6m. Acad. Sciences," 
xxi. p. 729) the quantity of heat required to raise a given 
mass of water from 0** to t"" C. is proportional to 

«+ -000 02^2 + -000 000 3^3. . . . (i) 

The mecm thermal capacity of a body between two stated 
temperatwres is the quantity of heat required to raise it 
from the lower of these temperatures to the higher, 
divided by the difference of the temperatures. The mean 
thermal capacity of a given mass of water between 0° C. 
and f is therefore proportional to 

1 + -000 02< + -000 000 3^2. ... (2) 



88 UNITS AND PHYSICAL CONSTANTS. [chap. 

The thermal capacity of a body at a started temperaJtwre 
is the limiting value of the mean thermal capacity as the 
range is indefinitely diminished. Hence the thermal 
capacity of a given mass of water at f is proportional to 
the differential coefficient of (1), that is to 

1 + OOO 04^ + -000 000 9<2. ... (3) 

Hence the thermal capacities at 0^ and 4* are as 1 to 
1*000174 nearly; and the thermal capacity at 0* is to 
the mean thermal capacity between 0** and 100"* as 1 to 
1-005. 

102. If we agree to adopt the capacity of unit mass oi 
water at a stated temperature as the unit of capacity, the 
unit of heat must be defined as n times the quantity of 
heat required to raise unit mass of water from this initial 

temperature through — of a degree when n is indefinitely 

n 

great. 

Supposing the standard temperature and the length of 
the degree of temperature to be fixed, the units both of 
heat and of thermal capacity vary directly as the unit of 
mass. 

In what follows, we adopt as the unit of heat (except 
where the contrary is stated) the heat required to raise 
a gramme of pure water through 1*" C. at a temperature 
intermediate between 0** and 4**. This specification is 
sufficiently precise for the statement of any thermal 
measurements hitherto made. 

103. The thermal capaciti/ of unit mass of a substance 
at any temperature is called the specific heat of the sub- 
stance at that temperature 



IX.] HEAT. 89 

Specific heat is of zero dimensions in length, mass, and 
i^ime. It is in fact the ratio 

increment of heat in the substance 
increment of heat in water 

ibr a given increment of temperature, the comparison 
being between eqital masses of the substance at the actual 
temperature and of water at the standard tempei^ture. 
"The numerical value of a given concrete specific heat 
merely depends upon the standard temperature at which 
the specific heat of water is called unity. 

104. The thermal capacity of unit volume of a sub- 
stance is another important element : we shall denote it 
by e. Let s denote the specific heat, and d the density of 
the substance ; then c is the thermal capacity of d units 
•of mass, and therefore c=^sd. The dimensions of c in 
lengthy mass, and time are the same as those of c?, namely, 

M 

Y^. Its numei'ical value will not be altered by any change 

in the units of length, mass, and time, which leaves the 
value of the density of water unchanged. 

In the O.G.S. system, since the density of water 
between 0° and 4" is very approximately unity, the 
thermal capacity of unit volume of a substance is the 
value of the ratio 

increment of heat in the substance 
increment of heat in water 
for a given increment of temperature, when the compari- 
json is between equal volumes, 

105. Mr. Herbert Tomlinson ("Proc. Roy. Soc," June 
19, 1885) has obtained the following deteiminations of 
specific heat from observations conducted in a uniform 



90 UNITS AND PHYSICAL CONSTANTS. [chap. 

manner with metallic wires well annealed. The wires 

were heated sometimes to 60*" C. and sometimes to 100*" 

C, and were plunged in water at 20**. The formulse are 

for the true specific heat at f C : — 

Aluminium, "20700 + •0002304< 

Copper, -09008 + -0000648^ 

German Silver, -09413 + -0000106^ 

Iron, -10601 + -000 140« 

Lead, 02998 + -00003U 

Platmum, -03198 + 'OOOOIS^ 

Platinum Silver, -04726 + •000028^ 

Silver, -05466 + -000044< 

Tin, -05231 + -000072* 

Zinc, -09009 + -000075< 

The formulse for the mean specific heat between 0* and t"* 
are obtained from these by leaving the first term un- 
changed and halving the second term. 

YioUe has made the following determinations of specific 
heat at f : — 

Platinum, -0317 + -000012* 

Iridium, -0317 + -000012* 

Palladium, -0582+ -000020* 

H. F. Weber has determined the specific heat of 
diamond to be 

•0947 + -000 994* - -000 000 30*2, 
and consequently the mean specific heat of diamond from 
0** to *• to be 

•0947 + -000 497* - -000 000 12*2. 
The mean specific heat of ice according to Kegnault is 
•504 between - 20* and 0^ and -474 between - 78** and 0'. 

106. The following list of specific heats of elementary 
substances is condensed from that given in Landolt and 
Bomstein's tables : — 



IX.] HEAT. 

Substance. Temperature. 

Aluminium, 16'to 97" 

Antimony, 13 ,, 106 

Arsenic (crystalline), 21,, 68 

„ (amorphous), 21 ,, 65 

Bismuth, 9 „ 102 

Borax (crystalline), „ 100 

,, (amorphous), 18,, 48 

Bromine, solid, -78 ,,-20 

,, liquid, 13 „ 45 

Cadmium, 0,, 100 

Calcium, 0„ 100 

Carbon, diamond, 11 

» graphite, 11 

,, wood charcoal,.. to 99 

Cobalt, 9 „ 97 

Copper, 15 „ 100 

Gold, „ 100 

Iodine, 9 „ 98 

Iridium, „ 100 

Iron, 60 

Lead, 19 to 48 

Lithium, 27 „ 99 

Magnesium, 20 ,, 51 

Manganese, 14 ,, 97 

Mercury, solid, -78 ,,-40 

„ liquid, 17 „ 48 

Molybdenum, 6 ,, 15 

Nickel, 14 „ 97 

Osmium, 19 „ 98 

Palladium, „ 100 

Phosphorus (yellow,8olid) -78 ,, 10 

( „ liquid) 49 „ 98 

(red), 15 „ 98 

Platinum, „ 100 

Potassium, -78 „ 



91 



Sp. Heat. 


Observer. 


•2122 


Regnault. 


•0486 


B^de. 


•0830 


J Bettendorflf & 
} WiUlner. 


•0758 


»» »» 


•0298 


B6de. 


•2518 


Mixter&Dana 


•254 


Kopp. 


•0843 


Regnault. 


•1071 


Andrews. 


•0548 


Bunsen. 


•1804 


it 


•1128 


H. F. Weber. 


•1604 




•1935 


>» 


•1067 


Regnault. 


•0933 


B6de. 


•0316 


VioUe. 


•0541 


Regnault. 


•0323 


VioUe. 


•1124 


Bystrom. 


•0315 


Kopp. 


•9408 


Regnault. 


•245 


Kopp. 


•1217 


Regnault. 


•0319 


>> 


•0335 


Kopp. 


•0659 


De la Rive and 
Marcet. 

• 


•1092 


Regnault. 


•0311 


»» 


•0592 


Violle. 


•1699 


Regnault. 


•2045 


Person. 


•1698 


Reg^ult. 


•0323 


Violle. 


•1655 


Regnault. 



92 UNITS AND PHYSICAL CONSTANTS. [chap. 

Sttbttanoe. Temperature. 8p. Heat. Otieerver. 

Rhodium, 10 „ 97 *0580 BQgnaalt 

Selenium, crystalline, 22 „ 62 -0840 j ^w^f * 

SiHcon, crystalline, 22 '1697 H. F, Weber. 

Silver, OtolOO "0559 Bunsen. 

Sodium, -28 „ 6 "2934 Begnauli 

Sulphur (rhomb, cryst), 17 „ 45 *163 Kopp. 

„ (newly melted), 15 „ 97 '1844 Begnault. 

Tellurium, crystalline,.... 21 „ 51 *0475 Kopp. 

Thallium, 17 „ 100 "0335 Regnault. 

Tin, cast, „ 100 -0559 Bunsen. 

Zinc, „ 100 -0935 „ 

Svhatcmcea not Elementary, 

Brass (4 copper 1 tin), hard, 15''to98^ *0858 Begnault 

„ „ soft, 14 „ 98 '0862 „ 

Ice, -20 „ -504 „ 

107. The following determinations of specific heat oJ 

liquids are by Kegnault. We have omitted decixna 

figures after the fourth, as even the second figure ii 

•different with different observers : — 

Alcohol. Chloroform. Oil of Turpentine. 

Temp. Sp. Ht. Temp. 8p. Ht. Temp. Sp. Ht 

-20** -6063 -30* -2293 -20** 3842 

'5475 -2324 '4106 

40 '6479 30 '2354 40 '4538 

80 '7694 60 '2384 80 '4842 

120 '5019 

160 '5068 

Ether. Bisulphide of Carbon. 

Temp. Sp. Ht Temp. Sp. Ht. 

-30' '5113 -30'* '2303 

'5290 2352 

30 '5468 30 '2401 

Schiiller has found the specific heat of liquid benzine at ^ 

to be 

•37980+ -001 44^. 



IX.] 



HEAT. 



9a 



108. The following table (from Miller's ''Chemical 
Phyaics/' p. 308) contains the results of Kegnault's ex- 
periments on the specific heat of gases. The column 
headed " equal weights '' contains the specific Jisats in the 
sense in which we have defined that term. The column 
headed " equal volumes " gives the relative thermal capa- 
cities of equal volumes at the same pressure and tem- 
perature : — 

Thermal Capacities of Gases and Vapours at 

Cotistant Fress'ure, 



Gas or Vapour, 



Air, - 

Oxygen, - - 
Nitrogen,- 
Hydrogen, 
Chlorine, - 
Bromine, - 
Nitrous Oxide, 
Nitric Oxide, - 
Carbonic Oxide 
Carbonic \ 

Anhydride, ) 
Carbonic Di-\ 

sulphide, / 
Ammonia, 
Marsh Gas, 
defiant Gas, - 
Arsenious ^ 

Chloride, / 
Silicic Chloride 
Titanic 
Stannic 
Sulphurous \ 

-^oihydride,/ 



9> 



Equal 



Vola. 



•2375 
•2405 
•2368 
•2359 
•2964 
3040 
3447 
•2406 
2370 

3307 

4122 

•2996 
3277 
4106 

7013 

•7778 
8564 
8639 

341 



Weights. 



•2375 
•2176 
•2438 
3-4090 
•1210 
•0555 
•2262 
•2317 
•2450 

•2163 

•1569 

•5084 
•5929 
•4040 

•1122 

•1322 
•1290 
•0939 

•154 



Gas or Vapour. 



Hydrochloric \ 
Acid, - -j" 

Sulphuretted \ 
Hydrogen, / 

Steam, 

Alcohol, - 

Wood Spirit, - 

Ether, - - 

Ethyl Chloride, 
Bromide, 
Disul- \ 

dphide, j 
yanide, 
CMoroform, - 
Dutch Liquid, 
Acetic Ether, - 
Benzol, 
Acetone, - 
Oil of Turpen- 
tine, - 
Phosphorus 
Chloride 



Equal 



Vols. 



»> 



} 
,} 



•2352 

•2857 

•2989 
•7171 
•5063 
1 -2266 
•6096 
•7026 

1 -2466 

•8293 
•6461 
•7911 
1-2184 
1-0114 
-8341 

2-3776 
•6386 



Weij(ht8 



1842 

2432 

4805 

4534 

4580 I 

4796 

2738 

1896 

4008 

4261 
1566 
2293 
4008 
3754 
4125 

5061 
1347 



94 



UNITS AND PHYSICAL CONSTANTS. [chap. 



109. E. Wiedemann (** Pogg. Ann.," 1876, No. 1, p. 
39) has made the following determinations of the specific 
heats of gases : — 

Specific HeaL 

AtO'. 

Air, 0-2389 

Hydrogen, 3*410 

Carbonic Oxi<le, 0-2426 

Carbonic Acid 0*1952 

Ethyl, 0-3364 

Nitric Oxide, 0-1983 

Ammonia, 0*6009 

Multiplying the specific heat by the relative density, 
he obtains the following values of 

Thermal Capacity of Equal Volumes. 



At 100*. 


At 200*. 


Relative 
Density. 

1 

0-0692 


• • * 


• ■ • 

• • • 


• • • 


• • • 


0-967 


0-2169 


0-2387 


1*629 


0*4189 


0-5015 


0*9677 


0-2212 


0-2442 


1-5241 


0*5317 


0-5629 


0-5894 



At 100*. 



At 200'. 



At 0\ 

Air, 0*2389 

Hydrogen, 0*2359 

Carbonic Oxide, . . 0*2346 

Carbonic Acid,... 0*2985 

Kthyl, 0*3254 

Nitric Oxide, 0*3014 

Ammonia, 0*2952 

The same author ("Pogg. Ann.," 1877, New Series, 
vol. ii. p. 195) has made the following determinations of 
specific heats of vapours at temperature f : — 

Specific Heat. 



0*3316 


0-3650 


0-4052 


0-4851 


0-3362 


0-3712 


0-3134 


0-3318 



VaTxiui- ^^S'^ ^^ Temp. 

\ ap*)ui . jjj Experiments. 

Chlorofonu, -JO'-g to 189'*-8 

Bromic Ethyl,.. 27°-9 to 189^-5 

Benzine, 34°-l to 115*^1 

Acetone, 26" -2 to 179° '3 

Acetic Ether, ... 32*'-9 to 113''-4 

Ether, 26° -4 to 188° '8 



•1341 -f 0001354/ 
1354 + 003560^ 
•2237 + -0010228/ 
•2984 -f -0007738/ 
-2738 + -0008700^ 
•.S725 + 0008536/ 



IX.] 



HEAT. 



95 



Kegnault's determinations for the same vapours were 
as follows : — 



Vapour. 



Bange of ^ 
Temperature. 



Mean Specific Heat for this Bange. 



Chloroform, 117° to 228" 

Bromic Ethyl, ... 77" 7 to 1 96" '5 

Benzme, 116" to 218" 

Acetone, 129" to 233" 

Acetic Ether,. ... 115" to 219" 

Ether, 70" to 220" 



According to 
Regnault. 

•1567 
•1896 
•3754 
•4125 
•4008 
•4797 



According to 
Wiedemann. 

•1573 
•1841 
•3946 
•3946 
•4190 
•4943 



Regnault has also determined the mean specific heat of 
bisulphide of carbon vapour between 80** and 147* to be 
•1534, and between 80' and 229* to be -1613. 

Melting Points and Heat op Liquefaction. 

110. VioUe has made the following determinations of 
melting points (" Com. Ren.," Ixxxix. p. 702) : — 



Silver, 954'* 

Gold, 1045 

Copper, 1054 



Palladium, 1500' 

Platinum, 1775 

Iridium, 1950 



This last temperature 1950* is very near to that of the 
hottest part of the oxyhydrogen flame. 

The same observer has found the latent heat of lique- 
fJEiction of platinum to be 27*2, and of palladium 36*3 
("Com. Ren," Ixxxv. p. 543, and Ixxxvii. p. 981). 

111. The following approximate table of melting points 
is based on that given in the second supplement to Watt's 
" Dictionary of Chemistry," pp. 242, 243 :— 



Platinum, 2000° 

Palladium, 1950 

Ck)ld, 1200 

Castlron, 1200 

Glass, 1100 



Copper, 1090° 

SUver, 1000 

Borax, 1000 

Antimony, 432 

Zinc, 360 



96 



UNITS AND PHYSICAL CONSTANTS. [chap. 



Lead, 330* 

Cadmimn, 320 

Bismuth, 265 

Tin, 230 

Selenium 217 

Cane Sugar, 100 

Sulphur, Ill 

Sodium,... 90 



Wax, 68** 

Potassium, 58 

Paraffin, 54 

Spermaceti, 44 

Phosphorus, 43 

Water,* 

Bromine, -21 

Mercury, -40 



Melting Latent 
Point. Heat. 

Silver, 1000* 211 

Zinc, 433 281 

Chloride of Calcium 

(CaC1.3H20),.... 28-6 40-7 

Nitrate of Potas- 
sium, 339 47-4 

Nitrate of Sodium, 310*5 63*0 



112. The following table (from Watt's "Dictionary of 
Chemistry," vol. iii. p. 77) exhibits the latent heats of 
fluidity of certain substances, together with their melting 
points : — 

Melting Latent 
Point. Heat. 

Mercury, -39" 2-82 

Phosphorus, . . 44 5*0 

Lead, 332 5*4 

Sulphur, 115 9-4 

lodme, 107 117 

Bismuth, 270 12-6 

Cadmium, 320 13-6 

Tin, 235 14-25 

The latent heat of fluidity of water was found by 
Regnault, and by Provostaye and Desains, to be 79*. 
Bunsen, by means of his ice-calorimeter (** Pogg. Ann.," 
vol. cxli. p. 30), has obtained the value 80-025. He 
finds the specific gravity of ice to be '9167. 

113. Chandler Roberts and Wrightson have compared 
the densities of molten and solid metals by weighing a 
solid metal ball in a bath of molten metal either of tLe 
same or a different kind (" Phys. Soc," 1881, p. 195, and 
1882, p. 102). They find that "iron expands rapidly (as 
much as 6 per cent.) in cooling from the liquid to the 
plastic state, and then contracts 7 per cent, to solidity ; 
whereas bismuth appears to expand in cooling from the 
liquid to the solid state about 2-35 per cent." The 
following is their tabular statement of results : — 



IX.] 



HEA*r. 



97- 



MetaL 



Sp. Grav. 
ofSoUd. 



Bismuth, 9*82 

Copper, 8*8 

Lead, 11-4 

Tin, 7-5 

Zinc, 7-2 

SUver, 10-57 

Iron (No. 4 foundry, 
Cleveland), 



6-95 



8p. Grav. 
of Liquid. 

10055 
8-217 

10-37 
7-025 
6-48 
9-51 

6-88 



Percentage of change in 

volume trtntk. cold solid 

to liquid. 

Decrease of vol. 2*3 

Increase of vol. 7-1 

9-93 

6-76 

11-1 

11-2 

1-02 



114. Change of volume in melting, from Kopp^s experi- 
ments (Watt's " Die," Art. Heat, p. 78) :— 

Phosphorus, Calling the volume at O*" unity, the volume at the 
melting point (44**) is 1*017 in the solid, and 1-052 in the 
liquid, state. 

Svlphur. Volume at 0° being 1, volume at the melting point 
(115**) is 1-096 in the solid, and 1*150 m the liquid, state. 

Wax. Volume at 0* being 1, volume at melting point (64*) is 
1'161 in solid, and 1-166 in liquid, state. 

Stearic Acid, Volume at 0" being 1, volume at melting point 
(70°) is 1079 in soUd, and 1-198 in Uquid, state. 

Rost*8 Fusible Afetal (2 parts bismuth, 1 tin, 1 lead). Volume at 
0** being 1, volume at 59° is a maximum, and is 1-0027. 
Volume at melting point (between 95° and 98°) is greater 
in liquid than in solid state by 1 '55 per cent. 

115. The following table (from Millers "Chemical 
Physics," p. 344) exhibits the change of volume of certain 
substances in passing from the liquid to the vaporous 
condition under the pressure of one atmosphere : — 

1 volume of water yields 1696 volumes of vapour, 
alcohol 528 

ether 298 

oil of turpentine 193 
O 



)> 



»» 



»» 



n 



>» 



»> 



»» 



»» 



} ' 



98 UNITS AND PHYSICAL CONSTANTS. [chap. 

116. The following table of boiling points and heats of 
vajjorization, at atmospheric pressure, is condensed from 
Landolt and Bornstein, pp. 189, 190 : — 

BoUing Latent Heat of ob««rver 
Point. Vaporisation. "«Morver. 

Alcohol, 77*9 202-4 Andrews. 

Bisulphide of Carbon, 46*2 86*7 

Bromine, 58 46*6 ,, 

Ether, 34*9 90-4 „ 

Mercury, 350 62-0 Person. 

Oil of Turpentme, 159-3 74-0 Brix. 

Sulphur, 316 362-0 Person. 

Water, 100 535-9 Andrews. 

117. Kegnaiilt's approximate formula for what he calls 
'Hhe total heat of steam at ^°/' that is, for the heat 
required to raise unit mass of water from C to ^ in the 
liquid state and then convert it into steam at Cy is 

606-5 + 'Z^ht. 

If the specific heat of water were the same at all tempera- 
tures, this would give 

606-5 -•695« 

as the heat of evaporation at f. But since, according 
to Regnault, the heat required to raise the water from 
0° to € is 

t + 000 02^' + -000 000 3«3, 

the heat of evaporation will be the difference between 
this and the " total heat," that is, will be 

606-5 - •695« - -000 02«2 _ -000 DOO 3<3, 

which is accordingly the value adopted by Kegnault as 
the heat of evaporation of water at f, 

118. According to Regnault, the increase of pressure 
at constant volume, and increase of volume at constant 



IX.] HEAT. 99 

pressure, when the temperature increases from 0° to 100°, 
haTe the following values for the gases named : — 

Q At Constant At Constant 

Volume. Pressure. 

Hydrogen -3667 '3661 

Air, -3665 3670 

Nitrogen, 3668 

Carbonic Oxide, 3667 3669 

Carbonic Acid, 3688 '3710 

Nitrous Oxide, 3676 3719 

Sulphurous Acid, 3845 '3903 

Cyanogen, 3829 3877 

Jolly has obtained the following values for the coeffi- 
cient of increase of pressure at constant volume : — 

Air, 00366957 

Oxygen, 00367430 

Hydrogen, 00365620 

Nitrogen, 0036677 

Carbonic Acid, 0037060 

Nitrous Oxide, 0037067 

Mendelejeff and Kaiander have determined the co- 
efficient of expansion of air at constant pressure to be 
•0036843. 

119. Kegnault's results as to the departures from Boyle's 
law are given in the form — 

-^l|i=l±A(»»-l)±B(m-in 

Vj denoting the volume at the pi^essure P^, Vq the volume 
at atmospheric pressure Pq, and m the ratio ^. 

For air, the negative sign is prefixed to A and the posi- 
tive sign to B, and we have 

log A = 3-04351 20, 
log B = 5-2873751. 




100 UNITS AND PHYSICAL CONSTANTS. [chap. 

For nitrogen, the signs are the same as for air, and we 
have 

log A = 4-8399376, 

log B = 6-8476020. 

For carbonic acid, the negative sign is to be prefixed 
both to A and B, and we have 

log A = 3-9310399, 
log B = 6-8624721. 

For hydrogen, the positive sign is to be prefixed both to 
A and B, and we have 

log A = 4-7381 736, 
log B = 6-9250787. 

120. The following table, showing the maximum pres- 
sure of aqueous vapour at temperatures near the ordinary 
boiling point, is based on Eegnault's determinations, as 
revised by Moritz (Guyot's Tables, second edition, collec- 
tion D, table xxv.) : — 

Dynes per 
sq. cm. 

9-779 xlO» 
9-814 



Temperature. 


Centims. of 
Mercury 
at Paris. 


99 0" 


73-319 


99-1 


73-584 


99-2 


73-849 


99-3 


74-115 


99-4 


74-382 


99-5 


74-650 


99-6 


74-918 


99-7 


75-187 


99-8 


75-457 


99-9 


75-728 


100-0 


76-000 


1001 


76-273 


100-2 


76-546 


100-3 


76-820 



»» 
»f 
» J 



9-849 
9-885 
9-920 
9-956 
9 992 
1 0028 X W 
1-0064 „ 
1-0100 ,» 
1-0136 „ 
1-0173 „ 
1-0209 ,, 
1-0245 „ 



a.] 



HEAT. 



101 



Temperature. 


Centims. of 
Mercury 
at Paris. 


Dynes per 
sq. cm. 


100-4 


77-095 


1-0282x106 


100 6 


77-371 


1-0319 „ 


100-6 


77-647 


1-0356 „ 


100-7 


77-925 


1-0393 „ 


100-8 


78-203 


1-0430 „ 


100-9 


78-482 


1-0467 „ 


101-0 


78-762 


1-0605 „ 



121. Mdximum Pressure of Aqueous Vapowr at various 
temperatures, in dynes per sq, centim. 



-20** 

-15 


1236 

1866 


-10 


2790 


- 6 


4150 





6133 


5 

10 

15 


8710 

12220 

16930 


20 

25 

30 

40 


23190 

31400 

42050 

73200 



50' l-226xl0«^ 

60 1-985 „ 

80 4-729 „ 

100 1014x106 

120 1-988 „ 

140 3-626 „ 

160 6-210 „ 

180 1006x107 

200 1-560 



Ifaadmum Pressure of various Vapours, in dynes per sq. cm. 





AlcoboL 


Ether. 


Sulphide of 
Carbon. 


Chloroform. 


-20** 


4455 


9-19 X W 


6-31 X 10* 




-10 


8630 


1-53 X W 


1 -058 X 10* 







16940 


2-46 „ 


1-706 „ 




10 


32310 


3-826 „ 


2-648 „ 




20 


59310 


5-772 „ 


3-975 „ 


2141 X W 


30 


1-048 X 10» 


8-468 „ 


5-799 „ 


3-301 „ 


40 


1-783 „ 


1-210 X 10« 


8-240 „ 


4927 „ 


50 


2-932 „ 


1-687 „ 


1-144 X 10« 


7-14 „ 


60 


4-671 „ 


2-301 „ 


1-554 „ 


1-007 X 10« 


80 


1 -084 X 106 


4 031 „ 


2-711 „ 


1-878 „ 


100 


2-265 „ 


6-608 „ 


4-435 „ 


3-24 „ 


120 


4-31 „ 


1 029 X 107 


6-87 „ 


5-24 „ 



102 



UNIT8 AND PHYSICAL CONSTANTS. [chap. 



122. The following are approximate values of the 
maximum pressure of aqueous vapour at various tempera- 
tures, in millimetres of mercury. They can be reduced 
to dynes per sq. cm. by multiplying by 133*4 : — 



O*' 


mm. 
4-6 


92^* 


mm. 
567 


112° 


mm. 
1150 


132° 


mm. 
2155 


10 


9-2 


94 


611 


114 


1228 


134 


2286 


20 


17-4 


96 


658 


116 


1311 


136 


2423 


30 


31-5 


98 


707 


118 


1399 


138 


2567 


40 


54-9 


100 


760 


120 


1491 


140 


2718 


50 


96-2 


102 


816 


122 


1588 


142 


2875 


60 


149 


104 


875 


124 


1691 


144 


3040 


70 


233 


106 


938 


126 


1798 


146 


3213 


80 


355 


lOS 


1004 


128 


1911 


148 


3393 


90 


525 


110 


1075 


130 


2030 


150 


3581 



123. The density (in gm. per cub. cm.) of aqueous vapour 
at any temperature t and any pressure jt? (dynesper sq. cm.), 
whether equal to or less than the maximum pressure, is 

•622 X -001276 p 

X ^ 



1 + •00366« 10«* 
If q denote the pressure in millimetres of mercury, the 
approximate formula is 

•622 X -001293 o 



1 X •00366< 760 

124. Temperature of evaporation and dew-point 
(Glaisher's Tables, second edition, page iv.). The fol- 
lowing are the factors by which it is necessary to mul- 
tiply the excess of the reading of the dry thermometer 
over that of the wet, to give the excess of the tempera- 
ture of the air above that of the dew-point : — 



Beading of 




Beading of 




Dry Bulb 
Therm. 


Factor. 


Dry Bulb 


Pacto 




Therm. 




-10°C.=14°F. 


8-76 


15°C. = 59°F. 


1-89 


- 5 23 


7-28 


20 68 


1-79 


32 


3-32 


25 77 


1-70 


+ 5 41 


2-26 


30 86 


1-65 


+ 10 50 


2-06 


35 95 


1-60 



IX.] HEAT. 103 

125. Critical temperatures of gases, above which thej 
cannot be liquefied (abridged from Landolt and Bomstein, 
p. 62) :— 

Critical Max. Pressure 

Temperature. of Gas at this Observer, 

o Temp. 

Hydrogen, - 174*2 98 '9 atm. Sarrau . 

Oxygen, -105-4 487 „ 

Nitrogen, -123*8 421 „ 

Carbonic Acid, 30*92 ,, Andrews. 

„ 32-0 77-0 „ Sarrau. 

Bisulphide of Carbon, 271*8 74*7 ,, Sajotschewsky. 

Sulphurous Acid, 155*4 78*9 ,, ,, 

Chloroform,.... 260*0 64*9 ,, ., 

Benzol, 280*6 49*5 ,, ,, 

Alcohol, 234*3 62*1 „ 

Ether, 1900 36*9 „ 

Conductivity. 

126. By the tliermal conductivity of a substance 
at a given temperature is meant the value of k in the 
expression 

q^kA^^:i^H, (1) 

X 

where Q denotes the quantity of heat that flows, in time 
t, through a plate of the substance of thickness Xy the area 
of each of the two opposite faces of the plate being A, 
and the temperatures of these faces being respectively 
i?, and V2» ®^ch differing but little from the given temper- 
ature. The lines of flow of heat are supposed to be 
normal to the faces, or, in other words, the isothermal 
Bui*faces within the plate are supposed to be parallel to 
the faces ; and the flow of heat is supposed to be steady, 
in other words, no part of the plate is to be gaining or 
losing heat on the whole. 



104 UNITS AND PHYSICAL CONSTANTS. [chap. 

Briefly, and subject to these understandings, conduc- 
tivity may be defined as the quantity of heat that passes 
in unit tiiney through unit area of a plate whose thickness 
is u/nityy when its opposite faces differ in temperature by 
one degree, 

127. Dimensions of Conductivity, From equation (1) 
we have 

]q— Q ^ /2) 

v^ - Ti * A< 

The dimensions of the factor — ^ — are simply M, since 

the unit of heat varies jointly as the unit of mass and 
the length of the degree. The dimensions of the factor 

- — are =r- ; hence the dimensions of k are --— . This is 
At LiT ij^ 

on the supposition that the unit of heat is the heat 

required to raise unit mxiss of water one degree. In 

calculations relating to conductivity it is perhaps more 

usual to adopt as the unit of heat the heat required to 

raise unit volume of water one degree. The dimensions of 

Q . . . . L2 

will then be L^, and the dimensions of h will be 7p • 



i?2 - 1? ' T 

These conclusions may be otherwise expressed by say- 

M 
ing that the dimensions of conductivity are =r-= when the 

liT 
thermal capacity of unit mass of water is taken as unity, 

and are — when the capacity of unit volume of water is 

taken as unity. In the C.G.S. system the capacities of 
unit mass and unit volume of water are practically 
identical. 



IX.] HEAT. 105 

n 

128. Let c denote the thermal capacity of unit volume 
of a substance through which heat is being conducted. 

Then - denotes a quantity whose value it is often neces- 
sary to discuss in investigations relating to the transmis- 
sion of heat. We have, from equation (2), 

c v^-Vj^' At^ 

Ok 
where Q' denotes — . Hence - would be the numerical 

c c 

value of the conductivity of the substance, if the unit of 

heat employed were the heat required to raise unit volume 

of the substance one degree. Professor Clerk Maxwell 

k 
proposed to call - the thermometric conductivity, as dis- 

c 

tinguished from k the thermal or calorimetric conductivity. 

"We prefer, in accordance with Sir Wm. Thomson's article, 

k 
" Heat," in the Encychpcedia Britannica^ to call - the 

diffusimty of the substance for heat, a name which is 

k 
based on the analogy of - to a coefficient of diffusion. 

Coefficient of Diffusion, 

129. There is a close analogy between conduction and 
diffusion. Let x denote the distance between two 
parallel plane sections A and B to which the diffusion is 
perpendicular, and let these sections be maintained in 
constant states. Then, if we suppose one substance to be 
at rest, and another substance to be diffusing through it, 
the coefficient of diffusion K is defined by the equation 

y=4 (1) 



106 UNITS AND PHYSICAL CONSTANTS. [chap. 

where y denotes the thickness of a stratum of the mixture 
as it exists at B, which would be reduced to the state 
existing at A by the addition to it of the quantity which 
diffuses from A to B in the time t 

When the thing diffused is heat, the states at A and B are 
the temperatures v^ and v^ and y denotes the thickness of 
a stratum at the lower temperature which would be raised 
to the higher by the addition of as much heat as passes 
in the time t. This quantity of heat, for unit area, will be 

which must therefore be equal to 

yc{v^ - v^\ 

whence we have 

k t 

y = - -. 

c X 

k 
The " thermometric conductivity " - may therefore be re- 

c 

garded as the coefficient of diffusion of heat. 

130. When we are dealing with the mutual inter- 
diffusion of two liquids, or of two gases contained in a 
closed vessel, subject in both cases to the law that the 
volume of a mixture of the two substances is the sum of 
the volumes of its components at the same pressure, the 
quantity of one of the substances which passes any section 
in one direction must be equal (in volume) to the quantity 
of the other which passes it in the opposite direction, 
since the total volume on either side of the section 
remains unaltered ; and a similar equality must hold for 
the quantities which pass across the interval between 
two sections, provided that the absorption in the interval 



IX.] HKAT. 107 

itself is negligible. Let x as before denote the distance 
between two parallel plane sections A and B to which the* 
diffasion is perpendicular. Let the mixture at A consist 
of m parts by volume of the first substance to 1 - m of 
the second, and the mixture at B consist of n parts of 
the second to 1 - n of the first, m being greater than 1 - w, 
and therefore n greater than 1 -m. The first substance 
will then diffuse from A to B, and the second in equal 
quantity from B to A. Let each of these quantities be 
such as would form a stratum of thickness z (the vessel 
being supposed prismatic or cylindrical, and the sections 
considered being normal sections), then z will be propor- 
tional to 

• 

7n-(l-n) , , . ^ m + n-l 
^ -t, that IS to L 

X X 

and the coefficient of interdiffusion K is defined by the 
equation 

==K'?i+'*-ri< (2> 

X 

The numerical quantity m + 7i - 1 may be regarded as 
measuring the difference of states of the two sections 
A and B. 

If y now denote 4)he thickness of a stratum in the con-^ 
dition of B which would be reduced to the state existing 
at A by the abstraction of a thickness z of the second 
substance, and the addition of the same thickness of the 
first, we have {\-n)y-\-z as the expression for the 
quantity of the first substance in the stratum after the 
operation. This is to be equal to my. Hence we have 



108 



UNITS AND PHYSICAL CONSTANTS. [chap. 



And sabstituting for z its value in (2) we have finally 



y = K-, 



X 



(4) 



which is of the same form as equation (1), ^ now denoting 
the thickness of a stratum of the mixture as it exists at B, 
which would be reduced to the state existing at A by the 
addition to it of the quantity of one substance which 
diffuses from A to B in the time t^ and the removal from 
it of the quantity of the other substance which diffuses 
from B to A in the same time. 

131. The following values of K in terms of the centi- 
metre and second are given in Professor Clerk Maxwell's 
** Theory of Heat," 4th edition, p. 332, on the authority of 
Professor Loschmidt of Vienna. 



Coefficients of Interdiffu^cfn of Gases 



Carbonic Acid and Air, 

Hydrogen, 
Oxygen,.... 



1423 
5614 
1409 
1586 
1406 
0982 
7214 
1802 
6422 
4800 

k 
These may be compared with the value of - for air, 

which, according to Professor J. Stefan of Vienna, is '256. 
The value of k for air, according to the same authority, 
is 6*58 X 10"", and is independent of the pressure. Pro- 
fessor Maxwell, by a different method, calculates its value 
At 5-4 X lO-''. 



Marsh Gas, 

Carbonic Oxide, 
Nitrous Oxide,.. 

Oxygen and Hydrogen, 

,, ,, Carbonic Oxide, 

Carbonic Oxide and Hydrogen, 

Sulphurous Acide and Hydrogen,... 



IX.] 



HEAT. 



109 



Results of Experim&rUs on Condticiivity of Solids, 
132. Principal Forbes' results for the conductivity of 
iron (Stewart on Heat, p. 261, second edition) are ex- 
pressed in terms of the foot and minute, the cubic foot 
of water being the unit of thermal capacity. Hence the 
value of Forbes* unit of conductivity, when referred to 



C.G.S., is 



(30-48)' 
60 



or 15*48; and his results must be 



multiplied by 15*48 to reduce them to the C.G.S. scale. 
His observations were made on two square bars ; the side 
of the one being 1^ inch, and of the other an inch. The 
results when reduced to C.G.S. units are as follows : — 



Temp. 
Cent. 

25 

50 

75 
100 
125 
150 
175 
200 
225 
250 
276 



l^-inch bar. 1-incb bar. 

-207 1536 

-1912 1460 

*1771 -1399 

*1656 -1339 

-1567 1293 

*1496 1259 

-1446 *1231 

-1399 -1206 

-1366 *1183 

-1317 *1160 

-1279 -1140 

-1240 -1121 

133. Neumann's results ("Ann. de. Chim. 
185) must be multiplied by '000848 to reduce them to 
our scale. They then become as follows : — 

Copper, 1*108 

Brass, -302 

Zinc, -307 

Iron, •164 

German Silver, '109 

Ice '0057 



" vol. Ixvi. p. 



110 UNITS AND PHYSICAL CONSTANTS. [chap. 

In the same paper he gives for bhe following substances 

k k , 

the values of -3 or - ; that is, the quotient of conductivity 

by the thermal capacity of unit volume. These require 
the same reducing factor as the values of k^ and when 
reduced to our scale are as follows : — 

Values of -. 

Coal, •ooiie'' 

Melted Sulphur, -00142 

Ice, -0114 

Snow, -00356 

Frozen Mould, 00916 

Sandy Loam, -0136 

Granite (coarse), -0109 

Serpentine, -00594 

134. Sir W. Thomson's results, deduced from observa- 
tions of underground thermometers at three stations at 
Edinburgh ("Trans. R. S. E.," 1860, p. 426), are given in 
terms of the foot and second, the thermal capacity of a 
cubic foot of water being unity, and must be multiplied 
by (30-48)2 or 929 to reduce them to our scale. The 
following are the reduced results : — 

it, or /• 

Conductivity. ^- 

Trap-rock of Calton Hill, -00415 -00786 

Sand of experimental garden, -00262 '00872 

Sandstone of Craigleith Quarry, 01068 -0231 1 

My own result for the value of -- from the Greenwich 

underground thermometers ("Greenwich Observations," 
1860) is in terms of the French foot and the year. As 
a French foot is 32-5 centims., and a year is 31557000 
seconds the reducing factor is (32-5)2-^ 31557000; that 
is, 3-347 X 10-^ The result is a- 

c* 
Gravel of Greenwich Observatory Hill, -01249 



IX.] HEAT. Ill 

Professors Ayrton and Perry ("Phil. Mag.," April, 
1878) determined the conductivity of a Japanese building 
stone (porphyritic trachyte) to be '006 9. 



1 35. Angstrom, in " Pogg. Ann.," vols. cxiv. (1861) and 
cxviii (1863), employs as units the centimetre and the 
minute ; hence his results must be divided by 60. These 
results, as given at p. 294 of his second paper, will then 
stand as follows : — 

Value of -. 
c 

Copper, first specimen, 1-216 (1 - -00214 t) 

„ second specimen, 1 '163 (1 - -001519 t) 

Iron, -224 (1 - '002874 t) 

He adopts for c the values 

-84476 for copper ; -88620 for iron, 
and thus deduces the following values of k : — 

Conductivity. 
Copper, first specunen, 1 027 (1 - '00214 t) 

„ second specimen, -983 (1 - -001519 t) 

Iron •199(1-002874 

136. A Committee, consisting of Professors Herschel 
and Lebour, and Mr. J. F. Dunn, appointed by the British 
Association to determine the thermal conductivities of 
certain rocks, have obtained results from which the 
following selection has been made by Professor Herschel : — 

Substance. ^?P^"«^HT*^? ^ - 

C.G.S. Units. c 

Iron pyrites, more than -01 more than '01 70 

Rock salt, rough crystal, '0113 -0288 

Fluorspar, rough crystal, 00963 '0156 

Quartz, opaque crystal, and 

quartzites, '0080 to -0092 '0175 to 0190 

Siliciou88andstone8(8lightlywet), '00641 to -00854 '0130 to -0230 



112 UNITS AKD PHYSICAL CONSTANTS. [chap. 

8»bd.nc,. ^l^uJl!'' I 
Galena, rough crystal, inter- 
spersed with quartz, •00705 •0171 

Sandstone and hard grit, dry, ... -00545 to '00565 '0120 
Sandstone and hard grit, thor- 
oughly wet, -00590 to -00610 -0100 

Micaceous flagstone, along the 

cleavage, -00632 "OllO 

Micaceous flagstone, across cleav- 
age, -00441 -0087 

Slate, along cleavage, 00550 to -00650 0102 

Do., across cleavage, -00315 to '00360 -0057 

Granite, various specimen8,aboat -00510 to -00550 -OlOOto -012D 
Marbles, limestone, calcite, and 

compact dolomite -00470 to -00560 -0085 to -0095 

Red Serpentine (Cornwall), -00441 -0065 

Caen stone (building limestone), -00433 -0089 

Whinstone, trap rock, and mica 

schist, -00280 to '00480 -0055 to -0095 

Clay slate (Devonshire), -00272 -0053 

Tough clay (sun-dried), -00223 -0048 

Do., soft (with one-fourth 

of its weight of water), 00310 -0035 

Chalk, 00200 to -00330 0046 to -0059 

Calcareous sandstone (firestone), '00211 -0049 

Plate-glass German and English, '00198 to -00234 -00395 to ? 

German glass toughened, '001 85 -00395 

Heavy spar, opaque rough crystal, '00177 

i'ire-brick, -00174 0053 

Fine red brick, '00147 -0044 

Fine plaster of Paris, dry plate, -00120 '^^^^\ab t 

Do., thoroughly wet, 00160 -0025/ 

White sand, dry, -00093 -0032 

Do. , saturated with 

water, about 00700 -0120 about 

House coal and cannel coal, '00057 to -00113 *0012 to *0027 

Pumice stone, -00055 



IX.] HEAT. 113 

137. P6clet in " Annales de Chimie," s6r. 4, torn. ii. p. 
114 [1841], employs as the unit of conductivity the tmns- 
mission, in one second, through a plate a metre square 
and a millimetre thick, of as much heat as will raise a 
cubic decimetre (strictly a kilogramme) of water one 
degree. Formula (2) shows that the value of this conduc- 
tivity in the C.G.S. system, is 

'''' ±.; that is, ' 



1 10000' '100 

His results must accordingly be divided by 100. 

The same author published in 1853 a greatly extended 
series of observations, in a work entitled "Nouveaux 
documents relatifs aux chauffage et k la ventilation." In 
this series the conductivity which is adopted as unity is 
the transmission, in one hour, through a plate a metre 
square and a metre thick, of as much heat as will raise a 
kilogramme of water one degree. This conductivity, in 

C.G.S. units is 

1000 100 1 ., ^ . 1 

; that IS, - 



1 10000 3600' '360 

The results must therefore be divided by 360. Those of 
them which refer to metals appear to be much too small. 
The following are for badly conducting substances : — 

Density. Conductivity. 
Fir, across fibres, -48 -00026 

„ alongfibres, -48 '00047 

Walnut, across fibres, "00029 

,, alongfibres, -00048 

Oak, across fibres, '00059 

Cork, -22 -00029 

Caoutchouc, -00041 

Guttapercha, -00048 

Starch paste, 1017 '00118 

Glass, 2-44 -0021 

H 



114 



UNITS AND PHYSICAL CONSTANTS. [chap. 



Density. 
Glass, 2-66 

Sand, quartz, 1*47 

Brick, pounded, coarse-grained, . 1 '0 

„ passed through ( , .-« 

silk sieve,.... ( 

Fine brickdust, obtained by decan- \ I -- 
tation, j ^ °^ 

Chalk, powdered, slightly damp, '92 

washed and dried, '85 

washed, dried, and \ , ^o 
compressed, j^ "^ 

Potato-starch, '71 

Wood-ashes, '45 

Mahogany sawdust, '31 

Woodcharcoal,ordinary,powdered, '49 

Bakers' breeze, in powder, passed \ .qk 
through silk sieve, / 



>* 



i* 



•41 



Conductivity. 
•0024 

00075 

00039 

00046 

00039 

00030 
00024 

00029 

00027 
00018 
00018 
00022 

00019 



Ordinary wood charcoal in powder, j .^ 
passea through silk sieve, \ 

Coke, powdered, '77 

Iron filings, 2*05 

Biuoxide of manganese, 1 '46 

Woolly Substances, 

Cotton Wool, of all densities, 

Cotton swansdown (molleton de \ 
coton), of all densities j 

Calico, new, of all densities, 

Wool, carded, of all densities, ... 
Woollen swansdown (molleton de \ 

laine) of all densities, J 

Eider-down, 

Hempen cloth, new, '54 

„ old, '58 

Writing-paper, white, '85 

Grey paper, unsized,... '48 

138. In Professor George Forbes's paper on conductivity 
^" Proc. R. S. E.," February, 1873) the units are the centim. 
and the minute; hence his results must be divided by 60. 



000225 

00044 
00044 
00045 



000111 

000111 

000139 
000122 

000067 

000108 
000144 
000119 
000119 
000094 



IX.] 



HEAT. 



115 



In a letter dated March 4, 1884, to the author of this 
work, Professor Forbes remarks that the mean tempera- 
ture of the substances in these experiments was - 10°, and 
expresses the opinion that bad conductors (such as most 
of these substances) conduct worse at low than at high 
temperatures — an opinion which was suggested by the 
analogy of electrical insulators. His results reduced to 
C.G.S. are— 



00223 
00213 



00177 
00115 
00081 
00072 

000717 

0005 

000453 

000405 

000335 

0003 

*000088 



Kamptulikon, .... 
Vulcanized india- 
rubber, 

Horn, 

Beeswax, 

Felt,.... 

Vulcanite, 

Haircloth, 

Cotton- wool, divided, 
„ pressed, 

Flannel, 

Coarse linen, 

Quartz, along axis, 



} 



»» 

»> 



a 



Ice, along axis, 

Ice, perpendicular to\ 

axiB, / 

Black marble, 

White marble, 

Slat«, 

Snow, 

Cork, 

Glass, 

Pasteboard, 

Carbon, 

Eoofing-felt, 

Fir,- parallel to fibre. 
Fir, across fibre, and\ . 

along radius, / 

Boiler-cement, -000162 

Paraffin, -00014 

Sand, very fine, -000131 

Sawdust, -000123 

Professor Forbes quotes a paper by M. Lucien De la 

Rive ("Soc. de Ph. et d'Hist. Nat de Geneve," 1864) in 

which the following result is obtained for ice, 

Ice, -00230. 

M. De la Rive's experiments are described in " Annales 
de Chimie," ser. 4, tom. i. pp. 504-6. 

139. Dr. Robert Weber (** Bulletin, Soc. Sciences Nat. 
de Neufchatel," 188), has found the following conductivi- 
ties and surface emissivities for five specimens of rock 
from the St. Gothard tunnel : — 



Quartz,perpendicular \ 
to axis, J 



00011 

000 089 

000087 
000 087 
-000087 
000 0833 
0000402 
000 0433 
000 0335 
0000355 
000 0298 
000922 
00124 
00057 
00083 

0040 

0044 



116 UNITS AND PHYSICAL CONSTANTS. [chap. 

Specimen No. IQS.—MiccLceous Gneiss, 

Conductivity, -000917 + 0000044^ 

Emissivity, -000185 + '0000023^ 

Specific Heat, -1778 + -00042^ 

Specimen No. 114. — Mica Schist, 

Conductivity, '000733 + '000 010^ 

Emissivity, -000207 + 'OOOOOIB^ 

Specific Heat, -18000 + 00044^ 

Specimen No. 124. — Eurite. 

Conductivity, '000862 + •00016^ 

Emissivity, -000249 + '00000009^ 

Specific Heat, '1682 + 'OOOee 

Specimen No. 140. — Gneiss, 

Conductivity, -0014 + -000003^ 

Emissivity, -00026 + •0000008< 

- Specific Heat, -1463 + •0009< 

Specimen No. 146. — Micaceous Schist, 

Conductivity, '000952 + '000 009^ 

Emissivity, '000168 + '0000023^ 

Specific Heat, '1697 + '0006^ 

Conductivity of Liquids, 

140. The conductivity of water, according to experi- 
ments by Mr. J. T. Bottomley ("Phil. Trans." 1881, 
April 3), is '002, which is nearly the same as the con- 
ductivity of ice. (See 138.) 

141. H. F. Weber ("Sitz. kon. Preuss. Akad." 1885), 
has made the following determinations of conductivities 
of liquids at temperatures of from 9** to 15*" C. He em- 
ploys the centimetre, the gi-amme, and the minute as 
units: we have accordingly divided the original numbers 
by 60 to reduce to C.G.S. 



rx.] 



HEAT. 



117 



Conduc- 
tivity. 

Water, -00136 

Aniline, '000408 

Glycerine, '000670 

Ether, '000303 



Methyl Alcohol, , 
Ethyl Alcohol,... 
Propyl Alcohol, . 
Butyl Alcohol,.., 
Amyl Alcohol,... 



Ameisen Acid,..., 

Acetic Acid, 

Propion Acid,.... 
Butyric Acid, ...., 
Isobutyric Acid, . 
Valerian Acid,... 
Isovalerian Acid,, 
Isocapron Acid, . . , 



Methyl Acetate,. . 
Ethyl Formiate,.., 

Ethyl Acetate, 

Propyl Formiate, . 
Propyl Acetate, .., 
Methyl Butyrate,, 
Ethyl Butyrate, . . 
Methyl Valerate,. 
Ethyl Valerate,... 



000495 
000423 
000373 
000340 
000328 

000648 
000472 
000390 
000360 
000340 
000325 
000312 
000298 

000386 
000378 
000348 
000357 
000327 
000335 
000318 
000315 
000307 



Amyl Acetate, 



Chloro Benzol, 

Chloroform, 

Chloro Carbon, 

Propyl Chloride, .. 
Isobutyl Chloride,. 
Amyl Chloride, 



Bromo Benzol, 

Ethyl Bromide, 

Propyl Bromide, 

Isobutyl Bromide,... 
Amyl Bromide, 



Ethyl Iodide,. 
Propyl Iodide, 



Isobutyl Iodide,. 
Amyl Iodide,.... 



Benzol, 

Toluol, 

Cymol, 

Oil of Turpentine, .... 

Sulphuric Acid, 

Bisulphide of Carbon, 

Oil of Mustard, 

Ethyl Sulphide, 



Conduc- 
tivity. 
•000302 

000302 
000288 
000252 
000283 
000278 
000284 

000265 
000247 
000257 
000278 
000237 

000222 
000220 
000208 
000203 

000333 
000307 
000272 
000260 

000765 
000343 
000382 
000328 



In the original paper these numbere are compared with 
the thermal capacities of the liquids per unit of volume, 
and with the calculated mean distances between their 
molecules. It is found that conductivity, multiplied by 
mean distance, divided by capacity, is a nearly constant 
quantity for the members of any one of the above groups. 
Comparing one group with another, its most widely dif- 
ferent values are represented by 19 and 23, if we except 
the last group, for which its value is between 26 and 27. 

Amission and Surface Conduction. 
142. Mr. D. M'Farlane has published (" Proc. Roy. Soc." 



118 



UNITS AND PHYSICAL CONSTANTS. [chap. 



1872, p. 93) the results of experiments on the loss of heat 
from blackened and polished copper in air at atmospheric 
pressure. They need no reduction, the units employed 
being the centimetre, gramme, and second. The general 
result is expressed by the formulae 

X = -000238 + 3-06 X IQ-H - 2-6 x lO'^^^ 
for a blackened surface, and 

i»= -000168 + 1 -98 X 10-«^- 1-7 x IQ-^t:' 
for polished copper, x denoting the quantity of heat lost 
per second per square centim. of surface of the copper, 
per degree of diflference between its temperature and that 
of the walls of the enclosure. These latter were blackened 
internally, and were kept at a nearly constant temperature 
of 14° C. The air within the enclosure was kept moist 
by a saucer of water. The greatest difference of tempera- 
ture employed in the experiments (in other words, the 
highest value of t) was 50° or 60° C. 

The following table contains the values of x calculated 
from the above formulae, for every fifth degree, within the 
limits of the experiments : — 



Diflference of 


Value of X. 


Ratio. 


Temperarure. 


shed Surface. 


Blackened Surface. 


5° 


•00017S 


•000252 


•707 ! 


10 


•000186 


•000266 


•699 ' 


15 


•000193 


•000279 


•692 , 


20 


•000-201 


•000289 


•695 ' 


25 


•000207 


•000298 


•694 


30 


•000212 


•000306 


•693 


35 


•000217 


•090313 


•693 1 


40 


000220 


•000319 


•693 1 


45 


000223 


•000323 


•690 ; 


50 


000225 


•000326 


•690 


55 


000226 


•000328 


•690 


60 


000226 


•000328 


•690 



IX.] 



HEAT. 



119 



143. Professor Tait has published (" Proc. R. S. E." 
1869-70, p. 207) observations bv Mr. J. P. Nichol on the 
loss of heat from blackened and polished copper, in air, 
at three different pressures, the enclosure being blackened 
internally and surrounded by water at a temperature of 
approximately 8° C* Professor Tait's units are the grain- 
degree for heat, the square inch for area, and the hour for 
time. The rate of loss per unit of area is 

heat emitted 
area x time ' 

The grain-degree is "0648 gramme-degree. 
The square inch is 6*4516 square centims. 
The hour is 3600 seconds. 
Hence Professor Tait's unit rate of emission is 



•0648 



6-4516 X 3600 



= 2-79xlO-« 



of our units. Employing this reducing factor, Professor 
Tait's Table of Results will stand as follows : — 



Pressure 1 '014 x 10* [760 niillims. of mercury]. 



Blackeued. 
Temp. Cent. Less per sq. cm. 

^ per second. 

61-2 -01746 

50-2 -01360 

41-6 -01078 

34-4 -00860 

27-3 -00640 

20-5 -00455 



Bright. 
Temp. Cent. Loss per sq. cm. 

o per second. 

63-8 -00987 

57-1 -00862 

50-5 00736 

44-8 -00628 

40-5 -00562 

34-2 -00438 

29-6 -00378 

23-3 -00278 

18-6 -00210 



•This temperature is not stated in the " Proceedings" but has 
been communicated to me by Professor Tait. 



120 UNITS AND PHYSICAL CONSTANTS. [chap. 

Pressure 1*36 x 10* [102 millims. of mercury]. 



62-5 01298 

57-0 -01168 

53-2 -01048 

47-5 -00898 

43 -00791 

28-5 -00490 



67-8 -00492 

61-1 -00433 

65 -00383 

49-7 -00340 

44-9 -00302 

40-8 00268 



Pressure 1*33 x 10* [10 millims. of mercury]. 



62-6 -01182 

57-5 -01074 

54-2 -01003 



65 -00388 

60 -00355 

50 -00286 



41-7 -00726 40 -00219 

37-5 -00639 30 -00157 

34 00569 23-5 -00124 

27-5 00446 

24-2 -00391 I 

Mechanical Equivalent of Heat. 

144. The value originally deduced by Joule from his 
experiments on the stirring of water waa 772 foot-pounds 
of work (at Manchester) for as much heat as raises a 
pound of water through 1° Fahr. This is 1389-6 foot- 
pounds for a pound of water raised 1° C, or 1389-6 foot- 
grammes for a gramme of water raised 1° C. As a foot 
is 30-48 centims., and the value of g at IManchester is 
981-3, this is 13896 x 30-48 x 981-3 ergs per gramme- 
degree; that is, 4-156 x 10^ ergs per gramme- degree. 

A later determination by Joule (" Brit. Assoc. Report," 
1867, pt. i. p. 522, or "Reprint of Reports on Electrical 
Standards," p. 186) is 25187 foot-grain-second units of 
work per grain-degree Fahr. This is 45337 of the same 
units per grain-degree Centigrade, or 45337 foot-gramme- 
second units of work per gramme-degree Centigrade; 
that is to say, 

45337 X (30-48)2 = 4-212x10' 
ergs per gramme-degree Centigrade. 



IX.] HEAT. 121 

In view of the fact that the B. A. standard of electrical 
resistance employed in this determination is now known 
to be too small by about 1 '3 per cent., and that the cur- 
rent energy converted into heat was accordingly under- 
estimated to this extent, the result ought now to be in- 
creased by 1*3 per cent, which will make it 

4-267 X 107. 

At the meeting of the Royal Society, January, 1878 
(" Proceedings," vol. xxvii. p. 38), an account was given by 
Joule of experiments recently made by him with a view 
to increase the accuracy of the results given in his former 
paper. ("Phil. Trans.," 1850.) His latest result from 
the thermal effects of the friction of water, as announced 
at this meeting, is, that taking the unit of heat as that 
which can raise a pound of water, weighed in vacuo, from 
60* to 61° of the mercurial Fahrenheit thermometer; its 
mechanical equivalent, reduced to the sea-level at the 
latitude of Greenwich, is 772*55 foot-pounds. 

To reduce this to water at 0* C. we have to multiply 
by 1-00089,* giving 773*24 ft. lbs., and to reduce to ergs 
per gramme-degree Centigrade we have to multiply by 

981-17 X 30-48 x?. 

5 

The product is 4-1624 x 10^. 

145. Some of the best determinations by various experi- 
menters are given (in gravitation measure) in the following 
list, extracted from "Watts' Dictionary of Chemistry," 
Supplement 1872, p. 687. The value 429*3 in this list 
corresponds to 4-214 x 10^ ergs : — 

* This factor is found by giving t the value 15*8 (since the tem- 
perature 60*5 Fahr. is 15 '8 Cent.) in formula (3) of art 101. 



122 UNITS AND PHYSICAL CONSTANTS. [chap. 
Hini| 



i» 



)) 



»» 



Joule,... 



Violle,... 



432 Friction of water and brass. 

433 Escape of water under pressure. 

441*6 Specific heats of air. 

425''2 Crushing of lead. 

^Q.o 5 Heat produced by an electric 

I current. 

435-2 (copper).. . \ 

434*9 (aluminium) I Heat produced by induced cur- 

435-8 (tin) ( rents. 

437*4 (lead) ) 

Regnault, 437 Velocity of sound. 

We shall adopt 4*2 x 10*^ ergs as the equivalent of 
1 gramme-degree ; that is, employing J as usual to denote 
Joule^s equivalent, we have 

J = 4*2x 107 = 42 millions. 

146. Heat and Energy of Combination with Oxygen, 



1 gramme of 


Compound 
formed. 


Gramme- 
degrees of heat 
produced. 


Equivalent 

Energy, in 

erga. 


Hvdrofiren 


C02 
S02 

P206 

ZnO 
FeW 
Sn02 
CuO 
C02 
CO^andH^O 

>» 
)) 


34000 AF 
8000 A F 
2300 A F 
5747 A 
1301 A 
1576 A 
1233 A 
602 A 
2420 A 
13100 A F 
11900 A F 
6900 A F 


1-43x1012 
3-36x1011 
9-66 X 101® 
2-41 X 10" 
5-46x1010 
6-62x1010 
518 „ 
2-53 „ 
1-02x10" 
5-50 „ 
5-00 „ 
2-90 „ 


Carbon, 


Sulphur, 


Phosphorus, 

Zinc, 


Iron, 


Tin, 


-^ **^» • 

Copper, 


Carbonic oxide,.... 
Marsh-sras 


defiant gas, 

Alcohol, 





Combustion in Chlorine. 



Hydrogen, . 
Potassium, 

Zinc, 

Iron, 

Tin, 

Copper, .... 



HCl 
KCl 

ZnCP 
Fe2Cl« 
SnCl^ 
CuCP 



23000 F T 


9-66x10" 


2655 A 


1-12 „ 


1529 A 


6-42x1010 


1745 A 


7-33 „ 


1079 A 


4-53 „ 


961 A 


404 „ 



IX.] HEAT. 123 

The numbers in the last column are the products of the 
numbers in the preceding column hy 42 millions. 

The authorities for these determinations are indicated 
by the initial letters A (Andrews), F (Favre and Silber- 
mann), T (Thomsen). Where two initial letters are 
given, the number adopted is intermediate between those 
obtained by the two experimenters. 

147. Difierence between the two specific heats of a gas. 

Let 8^ denote the specific heat of a given gas at con- 
stant pressure, 

^2 the specific heat at constant volume, 

a the coefficient of expansion per degree. 

V the volume of 1 gramme of the gas in cubic centim. 
at pressure ^ dynes per square centim. 

"When a gramme of the gas is raised from 0** to 1° at 
the constant pressure ^^, the heat taken in is s^, the increase 
of volume is av, and the work done against external resist- 
ance is avp (ergs). This work is the equivalent of the 
difierence between s^ and Sg ; that is, we have 

«j - ^2 = ^» w^eie J = 4-2 X 207. 

For dry air at 0** the value of vp is 7*838 x 10^, and a 
is '003665. Hence we find 5j - Sg = '0684. The value of 
«p according to Regnault, is '2375. Hence the value of 
«2 is •1691. 

The value of ^~ ^ , or -f*, for dry air at 0° and a 

megadyne per square centim. is 

?i.7 *2 = :2^ = 8-728 X 20-^ : 
V 783-8 



] 24 UNITS AND PHYSICAL CONSTANTS. [chap. 

^nd this is also the value of J — ? for any other gas (at 

the same temperature and pressure) which has the same 
•coefficient of expansion. 

148. CJhange of freezing point due to change of pressure. 

Let the volume of the substance in the liquid state be 
to its volume in the solid state of 1 to 1 + 6. 

When unit volume in the liquid state solidifies under 
pressure P+p, the work done by the substance is the 
product of P +p by the increase of volume e, and is there- 
fore Te+pe, 

If it afterwards liquefies under pressure P, the work 
Hlone against the resistance of the substance is Pe ; and if 
the pressure be now increased to P +p, the substance will 
be in the same state as at first. 

Let T be the freezing temperature at pressure P, 

T + t the freezing temperature at pressure T +p, 
I the latent heat of liquefaction, 
d the density of the liquid. 

Then d is the mass of the substance, and Id is the heat 
taken in at the temperature of melting T. Hence, by 
thermodynamic principles, the heat converted into mecbani- 
•cal effect in the cycle of operations is 

TT273 • ^' 
But the mechanical effect is pe. Hence we have 

T+273 J 

I _ e(T+273) . ,„> 

~p Jld~ <^^ 



nc.] HEAT. 125 

— - is the lowering of the freezing-point for an additional 

pressure of a dyne per square centim.; and — x 10^ will 

be the lowering of the freezing point for each addi- 
tional atmosphere of 10^ dynes per square centim. 
For water we have 

e=-087, Z= 79-25, T = 0, rf=l, 

- 1 X 1 0« = ?f ^ l^ll = -007 14. 
p 42 X 79-25 

Formula (3) shows that — is opposite in sign to e. 

P 
Hence the freezing point will be raised by pressure if the 

substance contracts in solidifying. 

149. Change of temperature produced by adiabatic com- 
pression of a fluid ; that is, by compression under such 
circumstances that no heat enters or leaves the fluid. 

Let a cubic centim. of fluid at the initial temperature 
^* C. and pressure p dynes per square centim. be put 
through the cycle of operations represented by the annexed 
"indicator diagram," ABCD, where horizontal distance 
from left to right denotes increase of volume and perpen- 
dicular distance upwards increase of pressure. 

In AD let the pressure be constant 
and equal to p. 

In BC let the pressure be constant 
and equal to ;? + tt, tt being small. 

Let AB and CD be adiabatics, so near 
together that AD and BC are very small 
compared with the altitude of the figure 
which is TT. 




126 UNITS AND PHYSICAL CONSTANTS. [cjhap. 

The figure will be ultimately a parallelogram, so that 
the changes of volume AD and BC will be equal ; let their 
common value be called edt, e denoting the expansion per 
degree at constant pressure; dt will therefore be the 
difference of temperature between A and D, or between 
B and C. We suppose this difference to be very small 
compared with the difference of temperature between A 
and B or between C and D. 

The cycle is reversible ; let it be performed in the direc- 
tion ABCD. Then heat is taken in as the substance 
expands from B to C, and given out as it contracts from 
D to A. 

The work done by the substance in the cycle is equal 
to the area of the parallelogram, which, being the product 
of the base edt by the height tt, is Tredt, The heat given 
out in DA is Cdt, C denoting the thermal capacity of a 
cubic centim. of the substance at constant pressure; 

hence the " efficiency " is — , and this, by the rules of 

J O 

T 

Thermodynamics, must be equal to — ^ , where t de- 

notes the increase of temperature from A to B. Put T 
for the absolute temperature 273 + <, then we have 

T^Te 

where t is the increase of temperature produced by the 
increase tt of pressure. 

150. Resilience as affected by heat of compression. 

The expansion due to the increase of temperature t, 

TTre^ 
above calculated, is re ; that is, -=— ; and the ratio of 

J O 



IX.] HEAT. 127 



this expansion to the contraction — , which would be pro- 

duced at constant temperature (E denoting the resilience 

ETe^ 
of volume at constant temperature), is -^p, : 1- Putting 

m for , the resilience for adiabatic compression will be 

J \j 

E 

; or, if m is small, E (1 +m) ; and this value is to 



1 - w 

be used instead of E in calculating the change of volume 

due to sudden compression. 

The same formula expresses the value of Young's 
modulus of resilience, for sudden extension or compression 
of a solid in one direction, E now denoting the value of 
the modulus at constant temperature. 

Eocafth'pleB, 

For compression of water between 10** and 11° we have 
E = 2-1 X low, T = 283, e = -000 092, C = 1 ; 

hence ^^=-0012. 

For longitudinal extension of iron at 10** we have 
E= 1-96 X 10^^ T = 283, e= -000 0122, C = -109 x 77 ; 

hence ^^ = -00234. 

Thus the heat of compression increases the volume- 
resilience of water at this temperature by about \ per 
cent., and the longitudinal resilience of iron by about 
\ per cent. 



128 



UNITS AND PHYSICAL CONSTANTS. [chap. 



For dry air at 0° and a megadyne per square centim., 
we have 

E = 10«, T = 273, e - }--, C = -2375 x -001276, 



m=__ = .288. 



\—m 



= 1 404. 



151. Eocpansions of Volumes per degree Cent, {abridged 

from WaM " Dictionary of Chemistry,** Article Heat, pp. 

67, 68, 71). 

Glass -00002 to -00003 

Iron -000035 „ '000044 

Copper, -000052 ,, -000057 

Platmum, -000026 ,, -000029 

Lead, -000084 „ -000089 

Tin,....., -000058 „ -000069 

Zinc, -000087 „ 000090 

Gold -000044 „ -000047 

Brass, -000053,, -000056 

Silver, -000057 „ -000064 

Steel, -000032 „ -000042 

Cast Iron, about '000033 

These results are partly from direct observation, and 
partly calculated from observed linear expansion. 

Expansion of Mercury, according to Regnault ( Watts* 

^* Dictionary,'* p. 56). 



Temp. = t. 

o 

. 

10 . 

20 . 

30 . 

50 . 

70 . 

100 . 



Volume at t. 



Expansion per 
degi*ee at t". 



1-000000 -00017905 

1-001792 -00017950 

1-003590 -00018001 

1-005393 '00018051 

1-009013 -00018152 

1-012655 '00018253 

1-018153 '00018405 



The temperatures are by air-thermometer. 



IX.] HEAT. 129 

The formula adopted by the Bureau International des 
Poida et Meaures for the volume at f C. (derived from 
Regnault's results) is 

1 + •000181792«+ -000 000 000 175^2 

+ •000 000 000 035116^^. 

Expansion of Alcohol and Ether, according to Kojjp 
{Watts' '* Dictionary ," p. 62). 

Volume. 



Temp. Alcohol. Ether. 

6 1-0000 10000 

10 1-0105 10152 

20 10213 10312 

30 1-0324 10483 

40 1-0440 1-0667 



152. Collected Data for Dry Air, 

Expansion from 0° to 100° at const, pressure, as 1 to 1 '367 

or as 273 to 373 

Specific heat at constant pressure, '2375 

„ ,, at constant volume, -1691 

Pressure-height at 0° C, about 7*99 x 10* cm., 

or about 26210 ft. 

Standard barometric column, 76 cm. = 29 '922 inches. 

Standard pressure, 1033*3 gm. per sq. cm. 

or 14-7 lbs. per sq. inch. 

or 2117 lbs. „ foot. 

or 1 '0136 X 10^ dynes per sq. cm. 

Standard density, at 0" C, -001293 gm. per cub. cm. 

or -0807 lbs. per cub. foot. 

Standard bulkiness, 773 -3 cub . cm. per gm. 

or 12-39 cub. ft. per lb. 
I 



1 30 UNITS AND PHYSICAL CONSTANTS, [chap. ix. 

Di*y and Moist Air, 

Mass of 1 Cubic Metre in Qrammes. 



Temp.C. Dry Air. Saturated Air. slto^tion. 

6 1293-1 1290-2 4-9 

10 1247*3 12417 9*4 

20 1204-6 1194-3 17*1 

30 1164-8 1146-8 30*0 

40 1127-6 1097-2 507 

If A denote the density of dry air and W that of vapour at 

3 

saturation, the density of saturated air is A - - W, or more 

exactly A - 608 W. 



131 



CHAPTER X. 

MAGNETISM. 

153. The unit magnetic pole, or the pole of unit strength, 
is that which repels an equal pole at unit distance with 
unit force. In the C.G.S. system it is the pole which 
repels an equal pole, at the distance of 1 centimetre, with 
a force of 1 dyne. 

If P denote the strength of a pole, it will repel an equal 

p2 

pole at the distance L with the force — -. Hence we have 
the dimensional equations 

P2L-2 = force = MLT-2, p2 = ML^T'S, P = M^L^T"'; 
that is, the dimensions of a pole (or the dimensions of 

strength of pole) are M*L^'~\ 

154. The work required to move a pole P from one 
point to another is the product of P by the difference of 
the magnetic potentials of the two points. Hence the 
dimensions of magnetic potential are 

^^- = ML^-- . M ~ 4l ~ 'T = m4l4t-\ 

155. The vrUensity of a magnetic ^'eW is the force which 
a unit pole will experience when placed in it. Denoting 



132 UNITS AND PHYSICAL CONSTANTS. [chap. 

this intensity by I, the force on a pole P will be IP. 
Hence 

IP = force = MLT-^ I = MLT-2. M'4l-^T = M*L"4t'^ ; 
that is, the dimensions o^ field-intensity are M*L" ^T"^ 

156. The moment of a magnet is the product of the 
strength of either of its poles by the distance between 

them. Its dimensions are therefore LP; that is, M*LTr"\ 
Or, more rigorously, the moment of a magnet is a 
quantity which, when multiplied by the intensity of a 
uniform field, gives the couple which the magnet ex- 
periences when held with its axis perpendicular to the 
lines of force in this field. It is therefore the quotient of 

a couple ML^T-^by a field-intensity M^L"*!"^; that 

is, it is M^L^T"^ as before. 

157. If different portions be cut from a uniformly mag- 
netized substance, their moments will be simply as their 
volumes. Hence the intensity of m,ag7ietization of a uni- 
formly magnetized body is defined as the quotient of its 
moment by its volume. But we have 

moment ^ m^L^t-i . L"^ = M^L " ^T'^ 
volume 

Hence intensity of magnetization has the same dimensions 
as intensity oi field. When a magnetic substance (whether 
paramagnetic or diamagnetic) is placed in a magnetic 
field, it is magnetized by induction, and the ratio of the 
intensity of the magnetization . thus produced to the 
intensity of the field is called the " coeflicient of magnetic 



X.] MAGNETISM. 133 

induction," or " coeflficient of induced magnetization," or 
the "magnetic susceptibility" of the substance. For 
paramagnetic substances (such as iron, nickel, and cobalt) 
this coefficient is positive; for diamagnetic substances 
(such as bismuth), it is negative ; that is to say, the induced 
polarity is reversed, end for end, as compared with that 
of a paramagnetic substance placed in the same field. 

158. It has generally been stated that " magnetic sus- 
ceptibilty " is nearly independent of the intensity of the 
field so long as this intensity is much less than is required 
for saturation. But R. Shida found ("Proc. Roy. Soc," 
Nov., 1882), in the softest iron wire, a very rapid varia- 
tion of susceptibility at low intensities. Under the 
influence of the earth's vertical force at Glasgow, '545, 
the susceptibility had the very large value 734 when the 
wire was stretched by a weight, and 335 when the weight 
was off. 

Under a magnetizing force 2*35, the susceptibilities, 
with and without the weight, were 235 and 154. 

Saturation was obtained with a magnetizing force of 
80*7, which produced magnetizations 1390 and 1430, the 
susceptibilities being therefore 17*1 and 17*6. 

With pianoforte wire (steel), the susceptibilities were 
67*5 and 69*3 under the earth's vertical force, and 13*2 
when saturation was just attained, with a magnetizing 
force of 107*5. The magnetization at saturation was 
1420, being about the same as for soft iron wire. 

With a square bar of soft iron nearly 1 centim. square, 
the susceptibility diminished from 19, under a magnetizing 
force of 18*2, to 7*6, under a magnetizing force of 189, 
which just produced saturation. 



134 UNITS AND PHYSICAL CONSTANTS. [chap. 



\ 



Examples, 

1. To find the multiplier for reducing magnetic in- 
tensities from the foot-grain-second system to the C.G.S. 
system. 

The dimensions of the unit of intensity are M^L~^T'\ 
In the present case we have M = -0648, L = 30-48, T = 1, 
since a grain is *0648 gramme, and a foot is 30*48 centini. 

Hence M*L"iT~' = ^^11 = -04611; that is, the foot- 
grain-second unit of intensity is denoted by the number 
•0461 1 in the C.G.S. system. This number is accordingly 
the required multiplier. 

2. To find the multiplier for reducing intensities from 
the millimetre-milligramme-second system to the C.G.S. 
system, we have 



1000 10 



Hence - is the required multiplier. 



3. Gauss states (Taylor's " Scientific Memoirs," vol. ii. 
p. 225) that the magnetic moment of a steel bar-magnet, 
of one pound weight, was found by him to be 100877000 
millimetre-milligramme-second units. Find its moment 
in C.G.S. units. 

Here the value of the unit moment employed is, in 

terms of C.G.S. units, M^L^'T-^ where M is 10"^ L is 

10-^ and T is 1 ; that is, its value islO~*.10~^^ = 10"**. 
Hence the moment of the bar is 10087*7 C.G.S. units. 



X.] MAGNETISM. 135 

4. Find the mean intensity of magnetization of the bar, 
assuming its specific gravity to be 7*85, and assuming that 
the pound mentioned in the question is the pound avoir- 
dupois of 453*6 grammes. 

Its mass in grammes, divided by its density, will be 
its volume in cubic centimetres ; hence we have 

453-6 



7-65 



= 57*78 = volume of bar. 



X . .. n i.- i.- moment 10088 ^^a £* 

Intensity of magnetization = — =-=r=-=T» = 174*0. 

•^ ^ volume 57*78 

5. Kohlmusch states (" Physical Measurements," p. 195, 
English edition) that the maximum of permanent mag- 
netism, which very thin rods can retain is about 1000 
millimetre-milligramme-second units of moment for each 
milligramme of steel. Find the corresponding moment 
per gramme in C.G.S. units, and the corresponding in- 
tensity of magnetization. 

For the moment of a milligramme we have 1000 
X 10-4 =10-1. 

For the volume of a milligramme we have (7*85)-^ 
X 10-^, taking 7 '85 as the density of steel. 

Hence the moment per gramme is 10"^ x 10^= 100, and 
the intensity of magnetization is 100 x 7*85 = 785. 

6. The m^aximtim intensity of magnetization for speci- 
mens of iron, steel, nickel, and cobalt has been deter- 
mined by Professor Rowland ("Phil. Mag.," 1873, vol. 
xlvi. p. 157, and November, 1874) — that is to say, the 
limit to which their intensities of magnetization would 
approach, if they were employed as the cores of electro- 
magnets, and the strength of current and number of con- 
volutions of the coil were indefinitely increased. Professor 



136 UNITS AND PHYSICAL CONSTANTS. [chap. 

Rowland's fundamental units are the metre, gramme, and 
second ; hence his unit of intensity is — ^ of the C.G.S. 

unit. His values, reduced to C.G.S. units, are 

At 12''C. At 220'C. 

Iron and Steel, 1390 1360 

Nickel, 494 380 

Cobalt, 800 (?) 

7. Gauss states (loc. cit) that the magnetic moment of 
the earth, in millimetre-milligramme-second measure, is 

3-3092 R3, 
R denoting the earth's radius in millimetres. Reduce 
this value to C.G.S. 

Since R^ is of the dimensions of volume, the other 
factor, 3*3092, must be of the dimensions of intensity. 
Hence, employing the reducing factor 10"^ above found, 
we have '33092 as the corresponding factor for C.G.S. 
measure ; and the moment of the earth will be 

•33092 R3, 
R denoting the earth's radius in centimetres — that is 
6-37 X 108. 
We have 

•33092 X (6-37 x 108)3 := 8-55 x 1025 
for the eartKs magnetic moinent in C.G.S. units. 

8. From the above result, deduce the intensity of mag- 
netization of the earth regarded as a uniformly magnetized 
body. 

We have 

. . .. moment 8*55 x lO^^ ^-^^ 

''^''"'•*y =vd^ = 1-083 710^ ="^^^°- 

This is about of the intensity of magnetization of 

2200 -^ ° 



X.] MAGNETISM. 137 

Gauss's pound magnet; so that 2 '2 cubic decimetres of 
earth would be equivalent to 1 cubic centim. of strongly 
magnetized steel, if the observed effects of terrestrial mag- 
netism were due to uniform magnetization of the earth's 
substance. 

9. Gauss, in his papers on terrestrial magnetism, em- 
ploys two different units of intensity, and makes mention 
of a third as ** the unit in common use." The relation 
between them is pointed out in the passage above referred 
to. The total intensity at Gottingen, for the 19th of 
July, 1834, was 4*7414 when expressed in terms of one 
of these units — the millimetre-milligramme-second unit ; 
was 1357 when expressed in terms of the other unit em- 
ployed by Gauss, and 1*357 in terms of the "unit in 
common use." In C.G.S. measure it would be *47414. 

159. A first approximation to the distribution of mag- 
netic force over the earths s surface is obtained by assuming 
the earth to be uniformly magnetized, or, what is mathe- 
matically equivalent to this, by assuming the observed 
effects to be due to a small magnet at the earth's centre. 
The moment of the earth on the former supposition, or 
the moment of the small magnet on the latter, must be 

*33092 R^ 
R denoting the earth's radius in centims. The magnetic 
poles, on these suppositions, must be placed at 
77" 50' north lat., 296" 29' east long., 
and at 77" 60' south lat., 116" 29' east long. 

The intensity of the horizontal component of terrestrial 

magnetism, at a place distant A" from either of these 

poles, will be 

33092 sin A" : 



138 UNITS AND PHYSICAL CONSTANTS. [chap. 

the intensity of the vertical component will be 

'66184 cos A"; 
and the tangent of the dip will be 

2 cotan A°. 
The magnetic potential, on the same supposition^ will 
be 

•33092 ?L cos A', 

r being variable. (See Maxwell, " Electricity and Mag- 
netism," vol. ii. p. 8.) Gauss's approximate expression 
for the potential and intensity at an arbitrary point on the 
earth's suiface consists of four successive approximations, 
of which this is the first. 

160. According to " Airy on Magnetism," the place of 
greatest horizontal intensity is in lat. 0° long. 259** E., 
where the value is '3733 ; the place of greatest total in- 
tensity is in South Victoria, about 70° S., 160° E., where 
its value is '7898, and the place of least total intensity is 
near St. Helena, in lat. 16° S., long. 355° E., where its 
value is -2828. 

161. The following mean values of the magnetic ele- 
ments at Greenwich have been kindly furnished by the 
Astronomer Royal (Dec, 1885) : — 

West Declination, IS** 15' 0- (<- 1883) x T'SO, 

Horizontal Force, 0-1809 + (<- 1883) x -00018. 

Dip, 67° 3r-8-(^- 1883) X 1-39. 

Vertical Force, 0-4374 -U- 1883) x -00007. 

= Horizontal force x tan. dip. 

Each of these formulae gives the mean of the entire 
year t, 

162. According to J. E. H. Gordon ("Phil. Trans.," 
1877, with correction in "Proc. Roy. Soc," 1883, pp. 



X.] MAGNETISM. 139 

4, 5), the rotation of the plane of polarisation between two 
points, one centimetre apart, whose magnetic potentials 
(in C.G.S. measure) differ by unity, is (in circular mea- 
sure) 

1-52381x10-^ 

in bisulphide of carbon, for the principal green thallium 
ray, and is 

2-248 X 10-6 

in distilled water, for white light. 

Mr. Gordon infers from BecquereFs experiments ("Comp. 
Rend.," March 31, 1879) that it is about 

3 X 10-^ 
for coal gas. 

According to Lord Rayleigh (" Proc. Roy. Soc./' Dec. 
29, 1884), the rotation for sodium light in bisulphide of 
carbon at 18* C. is -04202 minute. This is 

1-22231 X 10-s 
in circular measure. 



140 



CHAPTER XI. 

ELECTRICITY. 
Electrqstatics, 

163. If q denote the numerical value of a quantity of 
•electricity in electrostatic measure, the mutual force be- 
tween two equal quantities q at the mutual distance I will 

be^,. .In the C.G.S. system the electrostatic unit of 

electricity is accordingly that quantity which would repel 
an equal quantity at the distance of 1 centim. with a force 
of 1 dyne. 

Since the dimensions of force are — , we have, as 

regards dimensions, 

q- ml , o mP iA^~^ 

72 = 72 ' whence ^ = —, q = m It . 

164. The work done in raising a quantity of electricity 
-q through a difference of potential v is qv. 

Hence we have 

v = '!^ = mPt'Km-k-h = m¥t~\ 
In the C.G.S. system the unit difference of potential is 



CHAP. XI.] ELECTRICITY. 141 

that difference through which a unit of electricity must be 
raised that the work done may be 1 erg. 

Or, we may define potential as the quotient of quantity 
of electricity by distance. This gives 

V = 7/i*A"^ . l-^ = m^^r\ as before. 

165. In the O.G.S. system the unit of potential is the 
potential due to unit quantity at the distance of 1 centim. 

The capacity of a conductor is the quotient of the 
quantity of electricity with which it is charged by the 
potential which this charge poduces in it Hence we 
have 

capacity = ? = m^I^r^ , m''h~h = l. 

The same conclusion might have been deduced from 
the fact that the capacity of an isolated spherical con- 
ductor is equal (in numerical value) to its radius. The 
C.G.S. unit of capacity is the capacity of an isolated 
sphere of 1 centim. radius. 

166. The numerical value of a ctirrent (or the strength 
of a current) is the quantity of electricity that passes in 
unit time. 

Hence the dimensions of current are ?; that is, m^l^t~'-. 

t' 

The O.G.S. unit of current is that current which con- 
veys the above defined unit of quantity in 1 second. 

167. The dimensions of resistance can be deduced from 
Ohm's law, which asserts that the resistance of a wire is 
the quotient of the difference of potential of its two ends, 
by the current which passes through it. Hence we have 

resistance = m^l^t '^ ,m~H~ ^f = I'^t, 



\ 



142 UNITS AND PHYSICAL CONSTANTS. [chap. 

Or, the resistance of a conductor is equal to the time 
required for the passage of a unit of electricity through it, 
when unit difference of potential is maintained between 
its ends. Hence 

resistance = ^5?i2ii:2^?H!!!^ = <. mi iir> .«»- ir ?«= rt. 

quantity 

168. As the force upon a quantity q of electricity, in a 
field of electrical force of intensity i, is iq^ we have 

The quantity here denoted by i is commonly called the 
" electrical force at a point." 

Electromagnetics, 

169. A current C (or a current of strength C) flowing 
along a circular arc, produces at the centre of the circle an 
intensity of magnetic field equal to C multiplied by length 
of arc divided by square of radius. Hence C divided by 
a length is equal to a field-intensity, or 

C = length X intensity = L . M^L " *T"' = l4m4t~\ 

170. The quantiti/ of electricity Q conveyed by a cur- 
rent is the product of the current by the time that it lasts. 

The dimensions of Q are therefore L^M* 

171. The work done in urging a quantity Q through a 
circuit, by an electromotive force E, is EQ ; and the work 
done in urging a quantity Q through a conductor, by 
means of a differeyice of potential E between its ends, is 
EQ. Hence the dimensions of electromotive force, and 

also the dimensions of potential, are ML-T"^ . L ~ ^M " ', or 



XI.] 



ELECTRICITY. 



143 



172. The capacity of a conductor is the quotient of 
quantity of electricity by potential. Its dimensions are 
therefore 

M*L* . M " ^L - 'P ; that is, L'^T*. 

173. Resistance is — ; its dimensions are therefore 

C 

m4l^T-» . M " *L " *T ; that is, LT~\ 

174. The following table exhibits the dimensions of 
each electrical element in the two systems, together with 
their ratios : — 



! 

1 


Dimensions in 

electrostatic 

system. 


Dimensions in 

electromagnetic 

system. 


Dimensions in E.S. 




Dimensions in E.M. 


Quantity, 


M*L*T-i 


m4l* 


LT-i 


Current, 


M*L^T-2 


M*L*T-i 


LT-i 


Capacity, 


L 


L-1T2 


L2X-2 


Potential and ) 
electromo- > 
tive force, ) 


U^Juh-^ 


mMt-2 


, L-iT 


Resistance, 


L-iT 


LT-i 


L-2T2 



175. The heat generated in time T by the passage of a 

current C through a wire of resistance R (when no other 

C^RT 
work is done by the current in the wire) is — = — gramme 

J 

degrees, J denoting 4*2 x 10^ ; and this is true whether C 

and R are expressed in electromagnetic or in electrostatic 

units. 



144 UNITS AND PHYSICAL CONSTANTS. [chap. 

Ratios of the two sets of Electric Units, 

176. Let us consider any general system of units based 
on 

a unit of length equal to L centims., 
a unit of mass equal to M grammes, 
a unit of time equal to T seconds. 

Then we shall have the electrostatic unit of quantity 
equal to 

M^L'T"! C.G.S. electrostatic units of quantity, 
and the electromagnetic unit of quantity equal to 

M'L^C.G.S. electromagnetic units of quantity. 

It is possible so to select L and T that the electrostatic 
unit of quantity shall be equal to the electromagnetic 

unit. We shall then have (dividing out by M^L^) 

LT~^ C.G.S. electrostatic units 

= 1 C.G.S. electromagnetic unit ; 
or the ratio of the C.G.S. electromagnetic unit to the 

C.G.S. electrostatic unit is — . 

Now — is clearly the value in centims. per second of 

that velocity which would be denoted by unity in the 
new system. This is a definite concrete velocity ; and its 
numerical value will always be equal to the ratio of the 
electromagnetic to the electrostatic unit of quantity, 
whatever units of length, mass, and time are employed. 

177. It will be observed that the ratio of the two units 
of quantity is the inverse ratio of their dimensions ; and 



XI.] ELECTRICITY. 145 

the same can be proved in the same way of the other 
four electrical elements. The last column of the above 
table shows that M does not enter into any of the 
ratios, and that L and T enter with equal and opposite 
indices, showing that all the ratios depend only on the 

velocity -. 

Thus, if the concrete velocity _ be a velocity of v 

centims. per second, the following relations will subsist 

between the C.G.S. units : — 

1 electromagnetic unit of quantity = v electrostatic units. 

1 „ „ current =1? „ 

1 „ „ capacity =i;2 „ 

V electromagnetic units of potential = 1 electrostatic unit. 

v^ „ „ resistance = 1 „ 

178. Weber and Kohlrausch, by an experimental 
comparison of the two units of quantity, determined the 
value of t; to be 

3*1074 X 10^^ centims. per second. 

Sir. W. Thomson, by an experimental comparison of the 
two units of potential, determined the value of v to be 

2-825 X IQio. 

Professor Clerk Maxwell, by an experiment in wliich 
an electrostatic attraction was balanced by an electro- 
dynamic repulsion, determined the value of v to be 

2-88 X IQio. 

Professors Ayr ton and PeiTy, by measuring the capacity 

K 



146 UNITS AND PHYSICAL CONSTANTS. [chap. 

of an air-condenser both electroraagnetically and statically 
("Nature," Aug. 29, 1878, p. 470), obtained the value 

2-98 X 1010. 

Professor J. J. Thomson (" Phil. Trans.," 188 3, June 21), 
by comparing the electrostatic and electromagnetic mea- 
sures of the capacity of a condenser, and employing Lord 
Rayleigh's latest value of the B.A. resistance coils, de- 
termined V to be 

2-963x1010. 

All these values agree closely with the velocity of light 
in vacuo, of which the best determinations are, some of 
them a little less, and some a little greater than 

3 X 1010. 

We shall adopt this round number as the value of v. 

179. The dimensions of the electric units are rather 
simpler when expressed in terms of length, density, and 
time. 

Putting D for density, we have M = L^D. Making 
this substitution for M, in the expressions above obtained 
(§ 1 74), we have the following results : — 

Electrostatic. Electromagnetic. 

Quantity, D*L*r-i D^U 

Current, D^L^T-^ D*L^-i 

Capacity, L L-*T* 

Potential, D^L^T"' D^L^T-' 

Resistance, L-^T LT-i 

It will be noted that the exponents of L and T in these 
expressions are free from fractions. 



XI.] 



ELECTRICITY. 



147 



Specific Inductive Capacity, 

180. The specific inductive capacity of an insulating 
substance is the ratio of the capacity of a condenser in 
which this substance is the dielectric to that of a conden- 
ser in other respects equal and similar in which air is the 
dielectric. It is of zero dimensions, and its value exceeils 
unity for all solid and liquid insulators. 

According to MaxwelPs electro-magnetic theory of light, 
the square root of the specific inductive capacity is equal 
to the index of refraction for the rays of longest wave- 
length. 

Messrs. Gibson and Barclay, by experiments performed 
in Sir W. Thomson's laboratory ("Phil. Trans.," 1871, 
p. 573), determined the specific inductive capacity of solid 
paraffin to be 1*977. 

Dr. J. Hopkinson (*' Phil. Trans.," 1877, p. 23) gives the 
following results of his experiments on difierent kinds of 
flint glass : — 



Kind of 
Flint Glass. 



Very light» ... 

Ligbt, 

Dense, 

Double extra ) 
dense, \ 





Specific 
Inductive 


Quotient 


Index of 


Density. 


by 


Refraction 




Capacity. 


Density. 


for D line. 


2-87 


6-57 


2-29 


1-641 


3-2 


6-85 


214 


1-574 


3-66 


7-4 


202 


1-622 


4-5 


101 


2-25 


1-710 

1 
1 



In a later series of experiments (" Phil. Trans.," 1881, 
Dec. 16), Dr. Hopkinson obtains the following mean 
determinations : — 



148 UNITS AND PHYSICAL CONSTANTS. [chap. 

Specific Specific 

Glass. Inductive Density. Inductive 

Cai)acity. Capacity. 

Hard crown, 6*96 2*485 Paraflfin, 2*29 

A^ery light flint, 6*61 2*87 

Light flint, 672 3*2 

Dense flint, 7*38 3*66 

Double extra-dense flint, 9*90 4*5 

Plate, 8-45 

181. For liquids Dr. Hopkinson ("Proc. Roy. Soc," 
Jan. 27, 1881) gives the following values of /x^ (computed) 
and K (observed), K denoting the specific inductive 
capacity and /x^ the index of refraction for very long waves 
deduced by the formula 

where 6 is a constant. 

Petroleum spirit (Field's), 1 *922 1 -92 

Petroleum oil (Field's), 2*075 2*07 

,, (common), 2*078 2*10 

Ozokerit lubricating oil (Field's) 2086 2*13 

Turpentine (commercial), 2*128 2*23 

Castor oil, 2*153 4*78 

Sperm oil, 2*135 3*02 

Olive oil, 2*131 3*16 

Neatsfoot oil, 2*125 3*07 

This list shows that the equality of /tx^ to K (which 
Maxwell's theory requires) holds nearly true for hydro- 
carbons, but not for animal and vegetable oils. 

182. Wiillner (" Sitzungsber. konigl. bayer. Akad.," 

March, 1877) finds the following values of specific inductive 

capacity : — 

Paraffin, 1 *96 Shellac, ... 2 95 to 3 *73 

Ebonite 2*56 Sulphur, ... 2*88 to 3*21 

Plate glass, ... 6*10 



XI.] ELECTRICITY. 149 

Boltzmann ("Carl's Repertorium," x. 92—165) finds 

the following values : — 

Paraffin, 2*32 Colophonium, ... 2*55 

Ebonite, 3*16 Sulphur, 3*84 

SchUler ("Pogg. Ann.," clii. 535, 1874) finds:— 

Paraffin, ... 5 83 to 2*47 Caoutchouc, ... 2*12 to 2*34 
Ebonite, ... 2*21 to 2*76 Do., vulcanized, 2 '69 to 2*94 

Plate glass, 5*83 to 6*34 

Silow (" Pogg. Ann.," clvi. and clviii.) finds the following 

values for liquids : — 

Oil of turpentine, 2*155 to 2*221 

Benzene, 2*199 

Petroleum, 2*039 to 2*071 

Boltzmann (" Wien. Akad. Ber." (2), Ixx. 342, 1874) 
finds for sulphur in directions parallel to the three princi- 
pal axes, the values 

4*773, 3-970, 3-811. 
183. Quincke (" Sitz. Preuss. Akad.," Berlin, 1883j has 
made the following determinations. To explain the last 
two columns it is to be observed that, according to Max- 
well's theory, the charging of a condenser produces tension 
(or diminution of pressure) in the dielectric along the lines 
of force, and repulsion (or increase of pressure) perpen- 
dicular to the lines of force, the tension and the repulsion 
being each equal to 

K (A-B)2 
87rc2 

where K denotes the specific inductive capacity of the 
dielectric, c the distance between the two parallel plates 
of the condenser, and A - B their difference of potentials. 
Quincke observed the tension and repulsion, and computed 



150 



UNITS AND PHYSICAL CONSTANTS. 



[chap. 



K from each of them separately. The results are given 
in the last two columns, and are in every case greater 
than the " observed " value of K obtained in the usual 
way by comparison of capacities. 

The temperature printed below the index of refraction 
is the temperature at which the electrical experiments 
were performed. 



Ether, 

»i 

5 vols, ether to 1 bisul- 
phide of carbon, 

1 ether to 1 bisulphide, ... 

1 ether to 3 bisulphide, ... 

Sulphur in bisulphide of 
carbon (19 5 per cent.) 

Bisulphide of carbon from 
Kahlbaum, 

Bisulphide of carbon from 
Heidelberg, 

1 vol. bisulphide to 1 tur- 
pentine, 

Heavy benzol from ben- 
zoic acid, 

Pure benzol from benzoic 
acid, 

Light benzol, 

Rape oil, 

Oil of turpentine, 

Rock oil, 



Density 

tempera- 
tnre. 

1-7205 
atl4°-9 



•8134 
16° -4 

•9966 
16'* -6 

M360 
17"-4 

1 -3623 
12''-6 

1 -2760 
12° -2 

1 ^2796 
10° -2 

1-0620 

17°-8 

•8825 
15° -91 

•8822 
17°-64 

•7994 
17°^20 

•9159 
16°^4 

•8645 
17'1 

•8028 
17°0 



Index of 
refnetion 
and tem- 
perature. 

13605 
6° -60 

1-3594 
8° -37 

1-4044 
8° -50 

1-4955 
10°-50 

15677 
5°30 

1 6797 
8°^68 

1 0386 
7" 50 

1 -6342 
12°^98 

1 ^5442 
10°-92 

1 -5035 
13° -20 

1 •SOSO 
14° -40 

1^4535 
11° -60 

1 4743 
16°-41 

1 -4695 
16° -71 

1^4483 
16°^62 



Computed 



T 



Specific 

indnctlre /^ 

capacity from from 

obMired. tension. xepalBion . 

3-364 4-851 4672 



3-322 4623 4-660 



2871 4-136 4-392 



2-458 3 539 3 392 



2-396 3-132 3061 



2113 2-870 2-895 
2-217 2-669 2743 
1-970 2692 2^752 



1-962 2-453 2 540 



1-928 2-389 2^370 
2050 2325 2375 



r775 2-155 2-172 



2-443 2-385 3296 



1-940 2-259 2-356 



1-705 2138 2-149 



XI.] ELECTRICITY. 151 

184. Professors Ayrton and Perry have found the 
following values of the specific inductive capacities of 
gases, air being taken as the standard : — 



Air, 1-0000 

Vacuum, 0*9985 

Carbonic acid, ... 1 -0008 



Hydrogen, 0*9998 

Coal gas, 1-0004 

Sulphurous acid, 1 -0037 



Practical Units, 

185. The unit of resistance chiefly employed by practical 
electricians is the Ohrriy which is theoi*etically defined as 
10^ C.G.S. electro-magnetic units of resistance. 
The practical unit of electro-motive force is the Volty ' 
which is defined as 

10® C.G.S. electro-magnetic units of potential. 
The practical unit of current is the Ampere. It is de- 
fined as 

tV of the C.G.S. electro-magnetic unit current, 
or as the current produced by 1 volt through 1 ohm. 

The practical unit of quantity of electricity is the 
Coulomb, It is defined as 

tV of the C.G.S. electro-magnetic unit of quantity, 
or as the quantity conveyed by 1 ampere in 1 second. 

The practical unit of capacity is the Farad.* It is 
defined as 

10~® of the C.G.S. electro-magnetic unit of capacity, 
or as the capacity of a condenser which holds 1 coulomb 
when charged to 1 volt. 

* As the farad is much too large for practical convenience, its 
millionth part, called the microfarad, is practically employed, 
and condensers are in use having capacities of a microfarad and 
its decimal subdivisions. The microfarad is lO'^'^ of the C.G.S. 
electromagnetic unit of capacity. 



152 UNITS AND PHYSICAL CONSTANTS. [chap. 

The practical unit of work employed in connection with 
these is the Joule. It is defined as 

10^ ergs, 

or as the work done in 1 second by a current of 1 ampere 
in flowing through a resistance of 1 ohm. 

The corresponding practical unit of rate of working is 
the Watt. It is defined as 

10^ ergs per second, 

or as the rate at which work is done by 1 ampere flowing 
through 1 ohm. 

186. The standard resistance-coils originally issued in 
1865 as representing what is now called the ohm, were 
constructed under the direction of a Committee of the 
British Association, and their resistance was generally 
called the B. A. unit. The latest and best determinations 
by Lord Rayleigh and others have shown that it was 
about 1 — or, more exactly, 13 per cent. — too small, the 
actual resistance of the original B. A. coils being 

•987 X 109 c.G.S. 

187. An earlier unit in use among electricians was 
Siemens* unit, defined as the resistance at 0" C. of a 
column of pure mercury 1 metre long and 1 sq. millimetre 
in section. The resistance of such a column is about 

•943 X 109 c.G.S. 

The reciprocal of -943 is 1 06. 

188. The question of what electrical units should be 
adopted received great attention at the International 
Congress of Electricians at Paris in 1881 ; and the follow- 
ing resolutions were adopted : — 



XI.] ELECTRICITY. 153 

Resolutions adopted by the International Congress of 
Electricians at the sitting of Septemnher 22nd, 1881. 

1. For electrical measurements, the fundamental units, 
the centimetre (for length), the gramme (for mass), and 
the second (for time), are adopted. 

2. The ohm and the volt (for practical measures of 
resistance and electromotive force or potential) are to 
keep their existing definitions, 10^ for the ohm, and 10^ 
for the volt. 

3. The ohm is to be represented by a column of 
mercury of a square millimetre section at the temperature 
of zero centigrade. 

4. An International Commission is to be appointed to 
determine, for practical purposes, by fresh experiments, 
the length of a column of mercury of a square millimetre 
section which is to represent the ohm. 

5. The current produced by a volt through an ohm is 
to be called an ampere. 

6. The quantity of electricity given by an ampere in 
a second is to be called a coulomb. 

7. The ca{)acity defined by the condition that a coulomb 
charges it to the potential of a volt is to be called a farad. 

189. At a subsequent International Conference at Paris 
in 1884, it was agreed to define the ^^ legal ohm^^ as 
" the resistance of a column of mercury 106 centimetres 
long and 1 sq, millimetre in section, at the temperature of 
meltirig ice" 

The following summary of experimental results was 
laid before this Conference. The two columns of numerical 
values are inversely proportional, their common product 
being 100. One of them gives the value of Siemens' 



154 



UNITS AND PHYSICAL CONSTANTS. [chap. 



unit in terms of the theoretical ohm (10® C.O.S.), and 
the other gives the length of a column of pure mercury at 
0*" C, 1 square millimetre in section, which has a resist- 
ance of 1 theoretical ohm. 



Year. 


Obeerver. 


Siemenfl' 

Unit in 

Ohms. 


Colnmn of 

Mercury 

cm. 


Method. 


1864. 


British Assoc. Com., 


-9539 


104 83 


Brit. Association. 


1881. 


Rayleigh & Shuster, 


•9436 


105-98 


Do. 


1882. 


Rayleiffh, 


•9410 


106-28 


Do. 


1882. 


H.Weber, 


-9421 


10614 


Do. 


1874. 


Kohlrausch, 


•9442 


105-91 


Weber( 1 st method). 


1884. 


Mascart, 


•9406 


106 32 


Do. 


1884. 


Wiedemann, 


•9417 


106-19 


Do. 


1878. 


Rowland, 


•9453 
•9408 
9406 


105-79 
106 30 
106-32 


Kirch hoff. 


1882. 


Glazebrook, 


Do. 


1884. 


Mascart, 


Do. 


1884. 


F. Weber, 


•9400 


105-37 


Do. 


1884. 


R6iti, 


•9443 


105-90 


Rditi. 


1873. 


Lorenz, 


•9337 


107-10 


Lorenz. 


1884. 


Loreuz, 


•9417 


10619 


Do. 


1883. 


Rayleigh, 


•9412 


10624 


Do. 


1884. 


Zenz, 


•9422 


10613 


Do. 


1882. 


Doru, 


•9482 


105 46 


Weber (damping). 


1883. 


Wild, 


•9462 


105^68 


Do. 


1884. 


H. F. Weber, 


•9500 


105-26 


Do. 


1866. 


.foule, 


•9413 
•9430 


106-23 
106 04 


Joule. 




Mean, 





Several of the most distinguished physicists present 
expressed their opinion that 106^2 or 106^25 centimetres 
was the most probable value of the required length ; but 
in order to obtain unanimity it was agreed to adopt the 
length 106 centimetres, as above stated. 

190. By way of assisting the memory, it is useful to 
remark that the numerical value of the ohm is the same 



XI.] 



ELECTRICITY. 



155 



as the numerical value of a velocity of one earth-quadrant 
per second, since the length of a quadrant of the meridian 
is 10® centims. This equality will subsist whatever funda- 
mental units are employed, since the dimensions of resist- 
ance are the same as the dimensions of velocity. 

No special names have as yet been assigned to any 
electrostatic units. 

Electric SparL 

191. Sir W. Thomson has observed the length of spark 
between two parallel conducting surfaces maintained at 
known ditferences of potential, and has computed the 
corresponding intensities of electric force by dividing (in 
each case) the difference of potential by the^ distance, 
since the variation of potential per unit distance measured 
in any direction is always equal to the intensity of the 
force in that direction. His results, as given on page 258 
of " Papers on Electrostatics and Magnetism," form the 
first two columns of the following table : — 



DiBtance 
between 
Siirfaces. 


Intensity of 
force in 

Electrostatic 
Units. 


Difference of Potential between Surfaces. 


In Electrostatic 
Units. 


In Electromagnetic 
Units. 


•0086 
•0127 
•0127 
•0190 
•0281 
•0408 
•0663 
•0584 
•0688 
•0904 
•1056 
•1325 


267 1 
257-0 
262-2 
224-2 
200-6 
151-5 
1441 
139-6 
140-8 
134-9 
1321 
131-0 


2-30 

3-26 

3-33 

4-26 

5-64 

6-18 

8-11 

8-15 

9-69 

12-20 

13-95 

17-36 


6 90x1010 
9-78 „ 
9-99 ,. 
12-78 „ 
16-92 „ 
18-54 „ 
24-33 „ 
2445 „ 
29-07 „ 
36 60 „ 
41-85 „ 
£2-08 „ 



156 



UNITS AND PHYSICAL CONSTANTS. [chap. 



The numbers in the third column are the products of 
those in the first and second. The numbers in the 
fourth column are the products of those in the thii*d by 
3 X 1010. 

192. Dr. Warren De La Rue, and Dr. Hugo W. Miiller 
("Phil. Trans.," 1877) have measured the striking dis- 
tance between the terminals of a battery of choride of 
silver cells, the number of cells being sometimes as great 
as 11000, and the electromotive force of each being 1*03 
volt. Terminals of various forms were employed ; and 
the results obtained with parallel planes as terminals have 
been specially revised by Dr. De La Rue for the present 
work. These revised results (which were obtained by 
graphical projection of the actual observations on a larger 
scale than that employed for the Paper in the Philosophi- 
cal Transactions) are given below, together with the data 
from which they were deduced : — 

DATA. 



y 



No. of Cells. 


Striking Distance. 


In Inches. 


In Centims. 


1200 


0-012 


0305 


2400 


•021 


•0533 


3600 


•033 


•0838 


4800 


•049 


•1245 


5880 


•058 


•1473 


6960 


•073 1854 


8040 


•088 


•2236 


9540 


•110 


•2794 


11000 


•133 


•3378 



XI.] 



ELECTRICITY. 



157 



DEDUCTIONS. 



Electromotive 

Force in 

Volts. 


Striking 

Distance in 

Gentims. 


Volts per 
Gentim. 


1 

Intensity of Force 
In C.O.S. units. 

1 


Electromagnetic. 


Electro- 
static. 


1000 

2000 

3000 

4000 

5000 

6000 

7000 

8000 

9000 

10000 

11000 

11330 


•0205 
•0430 
•0660 
•0914 
•1176 
•1473 
•1800 
•2146 
•2495 
•2863 
•3245 
•3378 


48770 
46500 
45450 
43770 
42510 
40740 
38890 
37280 
36070 
34920 
33900 
33460 


4-88 X 
4-65 , 
455 
4-38 , 
4-25 
407 
3-89 . 
3-73 , 
3-61 
3-49 , 
3 39 
3-35 , 


10^2 


163 
155 
152 
146 
142 
136 
130 
124 
120 
116 
113 
112 



193. The resistance of a wire (or more generally of 
a prism or cylinder) of given material varies directly as 
its length, and inversely as its cross section. It is there- 
fore equal to 

■^ length 

section' 
where R is a coefficient depending only on the material. 
R is called the specific resistance of the material. Its 

reciprocal — is called the specific conductivity of the 
R 

material. 

R is obviously the resistance between two opposite 
faces of a unit cube of the substance. Hence in the C.G.S. 
system it is the resistance between two opposite faces of 
a cubic centim. (supposed to have the form of a cube). 

The dimensions of specific resistance are resistance x 
length ; that is, in electromagnetic measure, velocity x 
length ; that is, L^T'i. 



158 



UNITS AND PHYSICAL CONSTANTS. 



[chap. 



Resistance. 
194. The following table of specific resistances is 
altered from that given in former editions of this work 
by subtracting 1*88 per cent, from all the numbers in the 
column headed " Specific Resistance," this being the correc- 
tionrequired to reducetheresistanceof mercury from 96146, 
the value previously given, to 94340, which is the value 
resulting from the new definition of the " legalohm " : — 

Specific Resisla/aces in Electromagnetic Measure 
(ai 0° C, unless otherivise stated). 



1 


Specific 
Resistance. 


Percentage 

variation per 

degree at 

20' C. 


Specific 
Gravity. 


Silver, hard-drawn 


1579 

1611 

2114 

19474 

94340 

10781 
5-9x1018 

7-05xlO'o 
4^39 „ 
3-26 „ 
1-41 „ 
1-24 „ 
1-34 „ 

1-83 „ 
1-91 „ 

2-23x1016 
1 -36 X 10^5 
1-45x101* 
7-21x10" 
346 X 1023 
6-87 X 102* 


•377 
•388 
•365 
•387 
•072' 

•065 
100 

•47 
•47 
•653 
•799 
r259 
1-410 


10-50 
895 
19 27 
11-391 
13-595 

15-218 


CoDPer, ,, 


Gold, ,, 


Lead, pressed, 

Mercurv. liouid 


Gold 2, Silver 1, hard or]^ 
annealed, / 

Selenium at 100* C, crys-^ 
talline, / 

Water at 22*' C, 


„ with -2 per cent. H2SO4 

>» >» 0*0 ,, ,) 
20 

li 91 "" »» >» 
>> »» '*■*■ »> >» 

Sulphate of Zinc and Water \ 
ZnS04 + 23 H2O at 23" 0. , j 

Sulph.of Copper and VVater\ 
CUSO4 + 45 flaO at 22' C, / 

Glass at 200'' C, 


250' 


t1 «»w , 

300^ 


,, t^w , 

400" 
Gutta Percha aiik'^C.]....'. 

o-c, 



XI.] 



ELECTRICITY. 



159 



For the authorities on which this table is based see 
Maxwell, "Electricity and Magnetism," vol. i., last 
chapter. 

195. The following table of specific resistances of 
metals at 0° C. is reduced from Table IX. in Jenkin's 
Cantor Lectures. It is based on Matthiessen*s experi- 
ments. A deduction of 1 •SS per cent, has been made, as 
in the preceding table : — 



• 


Specific 
Resistance. 


Percentage 

of Variation 

for a degree 

at 20'* C. 


Resistance in 
Ohms of a 

Wire of 1 mm. 

diam. 

1 m. long. 


Silver, annealed, 

„ hard-drawn, 

CoDDer. annealed 


1492 

1620 

1584 

1620 

2041 

2077 

2889 

5581 

8982 

9638 

12358 

13103 

19468 

35209 

130098 

94340 

2419 
20763 
10779 


•377 
•388 
•365 

•365 

•365 

•387 
•389 
•354 
•072 

•031 
•044 
•065 


•0190 
•0206 
•0202 1 


,, hard-drawn, 

Gk>ld, annealed, 

., hard -drawn 


•0206 1 
•0260 
•0264 ' 


Aliuninium, annealed 

Zinc, pressed, 

Platinum, annealed 


•0368 1 
•0749 i 
•1144 i 


Iron, annealed, 

Nickel, annealed, 

Tin, pressed, 

Lead, pressed, 

Antimon v« nressed 


•1227 1 
•1573 , 
•1668 
•2479 
•4483 


Bismuth, pressed, 

Mercury, liquid, 

Alloy, 2 parts Platinum, 1 ] 
paurt Silver, by weight, j- 
hard or annealed, j 

German Silver, hard or an-"^ 
nealed, / 

Alloy, 2 parts Gold, 1 Sil- \ 
ver, by weight, hard or - 
annealed, 


1 6565 1 
r2012 ' 

•0308 
•2644 
•1372 






160 UNITS AND PHYSICAL CONSTANTS. [chap. 

Resistances of Condtictors of Telegraphic Cables per 

nautical mile, at 24° C, in CG.S. units. 

Red Sea, 7*79xl09 

Malta- Alexandria, mean, 3'42 

Persian Gulf, mean, 6*17 

Second Atlantic, mean, 4*19 

196. The following formulse are given by Benoit* for the 

ratio of the specific resistance at i° C. to that at 0° C. : — 

Aluminium, 1+003876^ 4-*00000l320^ 

Copper, l4--00367^ 4- -000 000 687^2 

Iron, 14--004516< + '000 006 828^2 

Magnesium, 1 + '003870^ + '000 000 863^2 

Silver, l + *003972< +'000000687^ 

Tin, l + -004028« + 000 005 826«* 

Mercury in glass tube, ' 



apparent resistance, not 
corrected for expansion, ^ 



1 + -0008649^ + -000 001 12^^ 



Adopting the formula 1+at for the ratio of the specific 
resistance at t" to that at 0**, MM. Cailletet and Bouty 
("Jour, de Phys.," July, 1885) have made the following 
determinations of the coefficient of variation a at very 
low temperatures : — 

Aluminium, 

Copper, 

Iron, 

Magnesium, 

Mercury, 

Silver, +30 to-102 '00385 

Tin, to- 85 '00424 

The new alloy called platinoid (consisting of German 
silver with a little tungsten) has been fouiid by Mr. J. T. 

* Benoit, *' Etudes exp^rimentales sur la Resistance ^lectrique 
sous rinfluence de la Temperature." Paris, 1873. 



Range of 
Temperature. 


CoefiScient of 
Variation. 


+ 28" to- 91" .. 


-00385 


-23 to-123 ... 


-00423 


to- 92 .. 


-0049 


to- 88 ... 


-00390 


-40 to- 92 .. 


-00407 



XI.] 



ELECTRICITY. 



161 



Bottoraley ("Proc. Roy. Soc.," May 7, 1885) to have an 
average variation of resistance with temperature of only 
•022 per cent, per degree centigrade, between 0° C. and 
100° C, being about half the variation of German silver. 
Its specific resistance ranges in different specimens from 
2-9 X 10-*^ to 3-7 X 10-» C.aS. 

Resistances of Liquids, 
197. The following tables of specific resistances of 
solutions are from the experiments of Ewing and Macgregor 
("Trans. Roy. Soc., Edin.," xxvii. 1873) :— 

Solutions at 10° C. Specific Resistance. 

Sulphate of Zinc, saturated, 3 '37 x 10^® 

,, „ minimum, 2*83 

Sulphate of Copper, saturated, 2 '93 

Sulphate of Potash, „ 1*66 

Bichromate of Potash, ,, 2*96 

The following table is for solutions of sulphate of cop- 
per of various strengths. The first column gives the 
ratio by weight of the crystals to the water in which they 
are dissolved : — 

strength, ^t^^'a 



»» 



)) 



»> 



>> 



1 to40 
30 
20 
10 

7 
5 



10167 
10216 
1 0318 
10622 
1 -0858 
1-1174 



Specific 
Resistance. 

16-44x1010 
13-48 

9-87 

5-90 

4-73 

3-81 






»> 



Strength. Density. 



1 to 4-146 1 

4 1 

3-297 1 

3 1 

2-5971 1 

saturated/ 



1386 
1432 
1679 
1823 



Specific 
Resistance. 

3-5 xlO^'' 
3-41 
3-17 
3 06 



>» 



J > 



2051 2-93 



>» 



»> 



The following table is for solutions of sulphate of zinc : — 



strength. Density. 



Ito40 
20 
10 

7 
5 



10140 
1-0278 
10540 
1 0760 
11019 



Specific 
Resistance. 

18-29x1010 
11-11 

6-38 

5-08 

4-21 



9) 



»> 



>' 



»5 



Strength. 



Density 
at 10°. 



lto3 1-1582 

2 1-2186 

1 -5 1 -270 

1 1 -3530 

saturated/ 



Specific 
Resistance. 

3-37x1010 
3 03 
2-85 
310 



»> 






)} 



162 



UNITS AND PHYSICAL CONSTANTS. [chap. 



The following table for dilute sulphuric acid is from 
Becker's experiments, as quoted by Jamin and Bouty, 
torn. iv. p. Ill : — 



Specific Resistance 



Density. 
110 
1-20 
1-25 
1-30 
1-40 
1-50 
1-60 
1-70 



/AtO\ 

1-37x1 010 

1 33 

1-31 

1-36 

1-69 

2-74 

4-82 

9-41 



At 8°. 
104 xlO'® 

•926 

•896 

•94 
r30 
213 
3 62 



)) 



*» 



»» 



5» 



»» 



S» 



At 16'. 

•845 X 10^0 

•666 

•624 

•662 
105 
1-72 



6-25 „ 



2-75 
423 






it 



»» 



)) 



At 24'. ^ 

•737 X 10^0 
•486 



•434 
•472 
•896 
1-52 
2-21 
307 



»» 
>» 
>» 
» 



Resistance of Carbons, 

198. The specific resistance of Carry's electric-light 
carbons at 20" C. is stated to be 

3-927 X 106 C.G.S., 

whence it follows that the resistance of a cylinder 1 metre 
long and 1 centimetre in diameter is just half an ohm. 
The specific resistance of Gau din's carbons is about 8*5 x ID** 



n 



)) 



retort carbon 



j> 



6-7 x 107 



)» 



J) 



graphite from 2^4 x 1 0^ to 4^2 x 10^ 
The resistance of carbon diminishes as the temperature 

J_ 
19 



increases, the diminution from 0° to 100° C. being -^^ for 



Carry's and — for Gaudin's. The resistance of an incan- 

descent lamp when heated as in actual use is about half 
its resistance cold. 



XI.] ELECTRICITY. 163 

Resistance of the Electric Arc, 

199. The difference of potentials between the two 
carbons of an arc lamp has been found by Ayrton and 
Perry ("PhU. Mag.," May, 1883) to be practically in- 
dependent of the strength of the current, when the dis- 
tance between them is kept constant. It was scarcely 
altered by tripling the strength of the current. The 
apparent resistance of the arc (including the effect of 
reverse electromotive force) is therefore inversely as the 
current. The difference of potentials was about 30 volts 
when the current was from 6 to 12 amperes. 

200. The following approximate determinations of the 
resistance of water and ice at different temperatures are 
contained in a paper by Professors Ayrton and Perry, 
dated March, 1877 (" Proc. Phys. Soc, London," vol. ii. 
p. 178):— 

Temp. Specific 

Cent. Resistance. 



o 



-12-4 2-240x1018 

- 6*2 1-023 „ 

- 5 02 9-486x1017 

- 3-5 6-428 „ 

- 3-0 5-693 „ 

2-46 4-844 „ 

- 1-5 3-876 „ 

- 0-2 2-840 „ 

+ 0-75 1-188 „ 

about + 2-2 2-48 x lO^e 

+ 4-0 9-1 xlO^s 

+ 7-75 5-4 xlO^^ 

+ 11-02 3-4 „ 

The values in the original are given in megohms, and 
we have assumed the megohm = 10^^ C.G.S. units. 



164 UNITS AND PHYSICAL CONSTANTS. [chap. 

According to F. Kohlrauscli (" Wied. Ann.," xxiv. 
p. 48, 1885) the resistance at 18° C. of water purified by 
distillation in vacuo is 4 x 10^^ times that of mercury. 
This makes its specific resistance 

3-76 X 1015. 

201. The specific resistance of glass of various kinds at 
various temperatures has been determined by Mr. Thomas 
Gray (" Proc Roy. Soc.," Jan. 12, 1882). The following 
are specimens of the results : — 

Bohemian Glass Tubing, density 2*43. 

At 60' 605x10^ At 160° 2*4 xW^ 

100 2 xlO^i 174 8-7 xlO^s 

130 2 X 1020 
Thomson's Electrometer Jar (flint glass), density 3*172. 

At 100" 2-06 X 1023 At 160** 2*45 x lO^i 

120 4-68x1022 180 5-6 x 102« 

140 1-06 „ 200 1-2 



,, *-\/V J. *. ,, 



The following are all at 60" C. : — 



Bohemian Beaker, 


4-25x1022 


density 2-427 


»» 19 


7-15 „ 


2-587 


Florence Flask, 


4-69 X 1020 


2-523 


Test Tube, 


1-44 „ 


2-435 


j» »> 


3-50 „ 


2-44 


Flint Glass Tube, 


3-89x1022 


2-753 


Thomson's Electro- ' 






meter Jar (flint 


- 1-02x1024 


3-172 


glass). 







202. The following appoximate values of the specific 
resistance of insulators after several minutes' electrifi- 
cation are given in a paper by Professors Ayrton and 
Perry ("Proc. Royal Society," March 21, 1878), "On 
the Viscosity of Dielectrics " : — 



XI.] 



ELECTRICITY. 



165 



Mica, 8*4xl(F 20 Ayrton and Perry. 

Gutta-Percha, 4-5xl0» 24 {«*^trmer1)ff '^ 

Shellac, O'OxlO^* 28 Ayrton and Perry. 

Hooper's Material, 1 *5 x lO^'^ 24 Recent cable tests. 

Ebonite, 2*8x1025 45 Ayrton and Perry. 

Paraffin, 3-4x102* 46 „ 

p J / Not yet measured with accuracy, but greater 

'** \ than any of tbe above. 

Air, Practically infinite. 

203. Particulars of Board of Trade Standard Gauge 
of Wires {Imperial Gauge) Nos. 4 to 20. 





Diameter. 




Resistance in ohms of 1 metre 








Sectional 
area. 

Sq. inchea 


length pure c 


opper at 0° C. 


No. 


MiUi- 
metres. 


Thou- 
sandths 
of inch. 






1 
Annealed. 


Hard-drawn. 

1 


4 


5-89 


232 


•04227 


•0005929 


•0006065 


5 


6-38 


212 


•03530 


7107 


7269 


6 


4-88 


192 


•02895 


8638 


8835 


7 


4-47 


176 


•02433 


•001029 


•001053 


8 


406 


160 


•02011 


1248 


1276 


9 


3-66 


144 


•01629 


1536 


1571 


10 


3-25 


128 


•012S7 


1948 


1992 


11 


2-95 


116 


-01057 


2364 


2418 


12 


2-64 


104 


•008494 


2951 


3019 


13 


2-34 


92 


•006647 


3757 


3842 


14 


203 


80 


•005026 


4992 


5106 


15 


1-83 


72 


•004070 


6142 


6283 


16 


1-63 


64 


•003216 


7742 


7919 


17 


1-42 


56 


•002463 


•01020 


; ^01043 


18 


1-22 


48 


•001809 


•01382 


1 -01414 


19 


1016 


40 


•001256 


•01993 


-02038 


20 


0-914 


36 


•000917 


•02462 


•02518 



166 



UNITS AND PHYSICAL CONSTANTS. Tchap. 



The heat generated per second in 1 metre length of 

— \ gm. deg.^ and 

(Q\2 
— 1 gm. deg., C. denoting the current 

in amperes, and D the diameter in millimetres. 

204. Resistance of 1 metre length of Wires of Imperial 
Gatige at 0** C, (For copper see preceding table.) 





Oerman Silver, 






No. 


Iron, 
annealed. an 
ha 


either 
nealed or 
rd*drawn. 


Platinum, 
annealed. 


Silver, annealed. 

1 


4 


•00.3606 


•007768 


•003361 


•0005583 


5 


4322 


9311 


4028 


6692 


6 


5253 


"01132 


4896 


8184 


7 


6261 


01349 


5836 


9694 


1 8 


7590 


01635 


7074 


•001175 ! 


i 9 


9339 


02012 


8705 


1446 1 


! 10 


•01184 


02552 


•01104 


1834 ; 


11 


•01438 1 


•03096 


•01340 


2226 


12 


•01795 


03867 


•01673 


2779 I 


13 


•02-285 


04922 


•02129 


3538 ' 


14 


•03036 


06540 


•02829 


4700 , 


15 


•03736 1 


•08047 , 


•03482 


5784 ■ 


16 


•04708 1 • 


1014 


•04388 


7290 ; 


17 


•06204 


1336 


•05782 


9606 


18 


•08405 


1811 


•07834 


•01301 


19 


•1212 


2611 


•1130 


•01876 


20 , 

1 


•1498 


3226 


•1396 


•02319 



Electromotive Force. 

205. The electromotive force of a Daniel I's cell was 
found by Sir W. Thomson (p. 245 of "Papers on Electricity 
and Magnetism ") to be 

•00374 electrostatic unit, 



XI.] ELECTRICITY. 167 

from observation of the attraction between two parallel 
discs connected with the opposite poles of a Danieirs 
battery. As 1 electrostatic unit is 3 x 10^^ electromag- 
netic units, this is -00374 x 3 x 10io= M22 x lO® electro- 
magnetic units, or 1*122 volt. 

According to Latimer Clark's experimental determina- 
tions communicated to the Society of Telegi-aph Engineers 
in January, 1873, the electromotive force of a DanielUs 
cell with pure metals and saturated solutions, at 64° F., 
is 1*105 volt, and the electromotive force of a Grove's 
cell 1 '97 volt. These must be diminished by 1 per cent, 
because they were deduced from the assumption that the 
B. A. unit of resistance was correct. They will thus be 
reduced to 1*094 and 1*95 volts. 

According to the determination of F. Kohlrausch 
("Pogg. Ann.," vol. cxli. [1870], and Erganz., vol. vi. 
[1874], p. 35) the electromotive force of a DanielFs cell 
is 1*138x108, and that of a Grove's cell 1*942x108. 
These must be diminished by 3 per cent., because they 
were deduced from the value '9717 x 10^ for Siemens' 
unit which is 3 per cent, too great. They will thus be 
reduced to 1*104 and 1*884 volts. 

H. S. Carhart ("Amer. Jour. Sci. Art.," Nov. 1884) 
has found the following different values for the electro- 
motive force of a Daniell's cell according to the strength 
of the zinc sulphate solution : — 



Per cent, 
of ZnSO^. 


Electromotive 
force in volts. 


Per cent. 


Electromotive 
force. 


1 


1*125 


10 


1*118 


3 


1*133 


15 


1*115 


5 


1*142 


20 


1111 


74 


1*1*20 


25 


1*111 



168 UNITS AND PHYSICAL CONSTANTS. [chap. 

He finds by the same method the electromotive force of 
Latimer Clark's standard cell to be 1434 volt. 

LordRayleigh ("Phil. Trans.," June 1884, p. 452) has 
determined the electromotive force of a Clark cell at 
15° C. to be 

1 -435 volt. 

The value formerly assigned to it was 1*457 volt, and 
was based on the assumption that the B. A. unit of resist- 
ance was correct. 

In a supplementary paper (Jan. 21, 1886) he gives the 
general result for any temperature T, 

1435{l-000077(«-15)}, 

together with full particulars as to the precautions neces- 
sary for securing constancy. 

206. Professors Ayrton and Perry have made deter- 
minations of the electromotive forces called out by the 
contacts, two and two, of a great number of substances 
measured inductively. The method of experimenting is 
described in the Proceedings of the Royal Society for 
March 21, 1878. The following abstract of their latest 
results was specially prepared for this work by Professor 
Ayrton in January, 1879 : — 



ELECTBICITY. 



JO ami] ai(i }« sjo^sjailuiax sSusAy 



i i 



i i i I 



3 ^ P I ? i 



— 3,1= 

= I11l 
Ivt 

^ III! 

ffl sill 

I iillH 

ili's' 



i ! iili 

i ^ i iflill 



CONTACT DIFFERENCES OP POTENTIAL IN VOLTS. 





J 


1 

5 


1 


1 


i 


P 


oi 


m™-'?. 

Dtatinedwatar, ..,.- 
Alum, saturated at: 


-Mi 
-■oas 


em 
-■m 

-108 
-070 
--173 


■Mi 
--006 


■171 

-■m 

-720 to 


-ZS6 
'34G 

--aw 

■OH 

i-eoo 

■872 


■177 


'A'S'.T»"'";} 

Sea Hit, Bpeclflc ' 






3 
i 

-i-l 
H\ 


rated at 1&''&C., .,( 
Zlno lulphate lolu- 

lDl»t11Icdwntern,iiedl 

M DtotiUed -water, 
1 stroug: HulphuMc 




10 DlBflil^ -«aier. 

6 Distilled -B3t6r, 
strong Bulphurlo 

e BtroDg eulphurle 


-■» 


•^SSlii 


Sulphuric ttcfd 

Kitricacid 

MercurouB Bulphate 






Distilled water, »itb ] 
atn^eofsmpliurio 













T^ average tempeni 
All tbe JJqulda and HlU employed n 
ff, were only coinmercliiUs pure. 



Solids fcith Liquids and Lvjuids with Liquids in Air, 



• 

a 


1 

o Amalgamated Zinc. 


n 


• 


1 

•2 
1 


AlumSolution.satu- 
rated at 16°^6 C. 


Copper Sulphate 
Solution, satu- 
rated at 16° C. 


Zinc Sulphate Solu- 
tion, Specific Grav- 
ity 1^125 at 16° '90. 


Zinc Sulphate Solu- 
tion, saturated at 
15°^3 C. 


1 Distilled Water, 
3 Zinc Sulphate. 


Strong Nitric Acid. 


-•105 

to 
+•156 


•231 


• • 


• • 


• • 


-•043 


■ • 


•164 


1 


-•686 


• ■ 


-•014 


















• • 


• • 


• a 


• • 


• • 


• • 


• t 


•090 








• • 


• • 


• • 


• • 


-•043 


■ • 


• • 


■ • 


•095 


•102 




-•666 


• • 


-•436 
















. 


-•637 




-•348 
















1 


-•288 






















• 
-•430 


-•284 


• • 


• • 


-•200 




-•095 










-•444 


■ • 


• • 


• • 


• • 


• • 


- -102 










-•344 






















■ • 


-•868 




















• • 


-•429 






















• • 


•016 


















■ « 


•848 


■ • 


• • 


1^298 


1^466 


1-269 




1^699 






• • 


• • 


• • 


•476 
















-•241 


• • 


• • 


• • 


• • 


t • 






• • 


• • 


•078 



Example of the above table :— Lead is poaitivo to d\&\iiil\&^'««.\x:;c^«s^^ 
the ooatact difference o/ potentials is O^lTl volt. 



172 UNITS AND PHYSICAL CONSTANTS. [chap. 

The authors point out that in all these experiments the 
unknown electromotive forces of cei*tain air contacts are 
included. 

From these tables we find we can build up the electro- 
motive forces of some well-known cells. For example, in 
a DanielFs cell there are four contact differences of potential 
to consider, and in a Grove's cell five, viz. : — 

DanielVa Cell. 

Volte. 
Copper and saturated copper sulphate, +0*070 

Saturated copper sulphate and saturated zinc sulphate, - 0*095 

Saturated zinc sulphate and zinc, +0*430 

Zinc and copper, +0*750 

M55 

Grovels Cell. 

Copper and platinum, +0*238 

Platinum and strong nitric acid, +0*672 

Strong nitric acid and very weak sulphuric acid, +0*078 

Very weak sulphuric acid and zinc, +0'241 

Zinc and copper, +0*750 

1*979 
TJiermoelectricity, 

207. The electromotive force of a thermoelectric circuit 
is called Thermoelectric force. It is proportional ccet. par. to 
the number of couples. The thermoelectric force of a single 
couple is in the majority of cases equal to the product of 
two factors, one being the difference of temperature of 
the two junctions, and the other the difference of the thermo- 
electric heights of the two metals at a temperature midway 
between those of the junctions. The current through the 
hot junction is from the lower to the higher metal when 
their heights are measured at the mean temperature. 



XI.] 



ELECTRICITY. 



173 



Our convention as to sign (that is, as to up and down 
in speaking of thermoelectric height) is the same as that 
adopted by Prof. Tait, and is opposite to that adopted in 
the first edition of this work. We have adopted it 
because it leads to the rule (for the Peltier and Thomson 
effects) that a current running down generates heat, and 
a current running up consumes heat. 

The following table of thermoelectric heights relative to 
lead can be employed when the mean temperature of the 
two junctions does not differ much from 19° or 20° 0. 
It is taken from Jenkin's " Electricity and Magnetism,'*^ 
p. 176, where it is described as being compiled from 
Matthiessen's experiments. We have reversed the signs 
to suit the above convention, and have multiplied by 100 
to reduce from microvolts to O.G.S. units. 



l^hermoelectric Heights at about 20° C. 



Bismuth, pressed comO _ q-qq 

mercial wire, / 

Bismuth, pure pressed \ _qqqq 

wire, / 

Bismuth, crystal, axial, - 6500 
, , equatorial ,....— 4500 

Cobalt, -2200 

German Silver, -1175 

- 41-8 


+ 10 
+ 10 
+ 90 
+ 120 



Quicksilver, 

Lead, 

Tin, 

Copper of Commerce, 

Platinum, 

Gold, 



Antimony, pressed wire + 

Silver, pure hard, + 

Zinc, pure pressed, + 

Copper, galvano-plas- \ 



280 
300 
370 

380 



600 



tically precipitated, j 
Antimony, pressed \ 
commercial wire, ...) 

Arsenic, + 1356 

Iron, pianoforte wire, + 1750 

Antimony, axial, + 2260 

,, equatorial, + 2640 

Phosphorus, red, + 2970 

Tellurium, +50200 

Selenium, + 8070O 



208. The following table is based upon Professor TaiVs 
thermoelectric diagram (" Trans. Roy. Soc, Edin.," vol. 
xxvii. 1873) joined with the assumption that a Grove's 
cell has electromotive force 1*97 x 10^ : — 



»» 5? 



174 UNITS AND PHYSICAL CONSTANTS. [chap. 

Thermoelectric Heifirhts at 
<• C. in C.G.S. units. 

Iron, + 1734- 4*87 « 

Steel, + 1139- 3*28^ 

Alloy, believed to be Platinum Iridium, + 839 at all temperatures. 

Alloy, Platinum 95; Iridium 5, + 622- '55 1 

90; „ 10, + 696- l-34« 

85; „ 15, + 709- "63^ 

M ,, 85; ,, 15, + 577 at all temperatures. 

Soft Platinum, - 61- 1*10^ 

Alloy, platinum and nickel, + 544- 1*10^ 

Hard Platinum, + 260- '75^ 

Magnesium, + 244- '95^ 

German Silver, _1207- 5-12^ 

Cadmium, + 266+ 4*29« 

Zinc, + 234+ 2-40« 

Silver, + 214+ l-50« 

Gold, + 283+ l'02t 

Copper, + 136+ '95t 

Lead, 

Tin, - 43+ '55t 

Aluminium, - 77+ '39^ 

Palladium, - 625- 3-59^ 

Nickel to HS^C, -2204- 5'\2t 

250" to 310° C, -8449 + 24-U 

from 340° C, - 307- 5-12< 

The lower limit of temperature for the table is - 18° C. 

for all the metals in the list. The upper limit is 416° C, 

with the following exceptions : — Cadmium, 258° C; Zinc, 

373'' C. ; German Silver, 175° C. 

Ex. 1. Required the electromotive force of a copper-iron 

couple, the temperatures of the junctions being 0* C. 

and 100° C. 

We have, for iron, + 1734 - 4-87< ; 

,, copper, + 136+ -95^; 

„ iron above copper, 1598-5*82^. 



5» 
J) 



XI. 1 ELECTRICITY. 175 

The electromotive force per degree is 

1598-5-82x50 = 1307, 
and the electromotive force of the couple is 

1307(100-0) = 130,700, 
tendiog from copper to iron through the hot junction. 

By the neutral point of two metals is meant the tem- 
perature at which their thermoelectric heights are equal. 
Ex. 2. To find the neutral point of copper and iron we 
have 

1598-5-82^ = 0, < = 275; 

that is, the neutral point is 275° 0. When the mean of 
the temperatures of the junctions is below this point, the 
current through the warmer junction is from copper to 
iron. The current ceases as the mean temperature attains 
the neutral point, and is reversed in passing it 

Ex. 3. F. Kohlrausch ('*Pogg. Ann. Erganz.," vol. vi. 
p. 35, 1874) states that, according to his determination, the 
electromotive force of a couple of iron and German silver 
is 24 X 10^ millimetre-milligramme-second units for 1° of 
difference of temperatures of the junctions, at moderate 
temperatures. Compare this result with the above Table 
at mean temperature 100". 

The dimensions of electromotive force are M^L'T"* ; 

hence the C.G.S. value of Kohlrausch's unit islO'^lO"' 
= 10"', giving 2400 as the electromotive force per degree 
of difference. 

From the above table we have 

Iron above German silver, 2941 + -25^, 

which, for t = 1 00, gives 2966 as the electromotive force 
per degree of difference. 



176 UNITS AND PHYSICAL CONSTANTS. [chap. 

Peltier and Thomson Effects. 

209. When a current is sent through a circuit com- 
posed of different metals, it produces in geneiul three 
distinct thermal effects. 

1. A generation of heat to the amount per second of 
C^R ergs, denoting the current, and R the resistance. 

2. A generation of heat or cold at the junctions. This 
is called the Peltier effect, and its amount per second in 
ergs at any one junction can be computed by multiplying 
the difference of thermoelectric heights at this junction 
by < + 273 and by the current, t denoting the centigrade 
temperature of the junction. If the current flows down 
(that is from greater to less thermoelectric height) the 
effect is a warming ; if it flows up, the effect is a cooling. 

Ex. 4. Let a unit current (or a current of 10 amperes) 
flow through a junction of copper and iron at 100* C. 
The thermoelectric heights at 100° C. are 

Iron, 1247 

Copper, 231 

Iron above copper, 1016 

Multiplying 1016 by 373, we have about 379,000 ergs, or 

— - of a gramme-degree, as the Peltier effect per second. 

Heat of this amount will be generated if the current is 
from iron to copper, and will be destroyed if the current 
is from copper to iron. 

3. A generation of heat or cold in portions of the cir- 
cuit consisting of a single metal in which tlie temperature 
varies from point to point. This is called the Thomson 
effect. Its amount per second, for any such portion of 



XI.] \ ELECTRICITY. 177 

the circuity is the difference of the thermoelectric heights 
of the twc ends of the portion, multiplied by 273 + <, 
i«rhere t denotes the half-sum of the centigrade tempera- 
tures of the ends, and by the strength of the current. 

The Thomson effect, like the Peltier effect, is reversed 
by reversing the current, and follows the same rule that 
heat is generated when the current is from greater to less 
thermoelectric height. 

Experiment shows that the Thomson effect is insensible 
in the case of lead; hence the thermoelectric height of 
lead must be sensibly the same at all temperatures. It is 
for this reason that lead is adopted, by common consent, 
as the zero from which thermoelectric heights are to be 
reckoned. 

Ex. 5. In an iron wire with ends at 0° C. and 100° C, 
the cold end is the higher (thermoelectrically) by 
4-87 X 100— that is, by 487. Multiplying this differ- 
ence by 273 + 1(0 + 100) or 323, we have 157300 as the 
Thomson effect per second for unit current. This amount 
of heat (in ergs) is generated in the iron when the current 
through it is from the cold to the hot end, and is destroyed 
when the current is from hot to cold. 

Ex. 6. In a copper wire with ends at 0° 0. and 100** 0., 
the hot end is the higher by '95 x 100 or 95. Multiply- 
ing this by 323, we have 30700 (ergs) as the Thomson 
effect per second per unit current. This amount of heat 
is generated in the copper when the current through it is 
from hot to cold, and destroyed when the current is from 
cold to hot. 

The effect of a current from hot to cold is opposite in 
these two metals, because the coefficients of ^ in the 

M 



/ 

y 



178 UNITS AND PHYSICAL CONSTANTS./ [chap. 

expressions for their thermoelectric heights (p. 174) have 
opposite signs. 

Relation between Thermoelectric Force and the Peltier and 

Thomson effects, 

210. The algebraic sum of the Peltier and ThomsoD 
effects (expressed in ergs) due to unit current for one second 
in a closed metallic circuit, is equal to the thermoelectric 
force of the circuit; and the direction of this thermoelectric 
force is the direction of a current round the circuit which 
would give an excess of destruction over generation of 
heat (so far as these two effects are concerned). 

Ex. 7. In . a copper-iron couple with junctions at 
0* C and 100* C, suppose a unit current to circulate in 
such a direction as to pass from copper to iron through 
the hot junction, and from iron to copper through the cold 
junction. 

The Peltier effect at the hot junction is a destruction 
of heat to the amount 1016 x 373 = 379,000 ergs. 

The Peltier effect at the cold junction is a generation of 
heat to the amount 1598 x 273 = 436,300 ergs. 

The Thomson effect in the iron is a destruction of heat 
to the amount 487 x 323 = 157,300 ergs. 

The Thomson effect in the copper is a destruction of 
heat to the amount 95 x 323 = 30,700 ergs. 

The total amount of destruction is 567,000, and of 
generation 436,300, giving upon the whole a destruction 
of 130,700 ergs. The electromotive force of the couple is 
therefore 130,700, and tends in the direction of the 
current here supposed. This agrees with the calculation 
in Example 1. " ^ ^ 



\ 



XI.] ELECTRICITY. 179 

Electrochemical Equivalents, 

211. The quantity of a given metal deposited in an 
electrolytic cell or dissolved in a battery cell (when there 
is no " local action ") depends on the quantity of electricity 
that passes, irrespective of the time occupied. Hence we 
can speak definitely of the quantity of the metal that is 
" equivalent to " a given quantity of electricity. By the 
electrochemical equivalent of a metal is meant the quantity 
of it that is equivalent to the unit quantity of electricity. 
In the C. G. S. system it is the number of grammes of 
the metal that are equivalent to the C.G.S. electromagnetic 
unit of electricity. 

Special attention has been paid to the electrochemical 

equivalent of silver, as this metal afibrds special facilities 

for accurate measurement of the deposit. The latest 

experiments of Lord Rayleigh and Kohlrausch agree in 

giving 

•01118 

as the C.G.S. electrochemical equivalent of silver.* 

The number of grammes of silver deposited by 1 
ampere in one hour is 

•01118 x^^^x 3600 = 4-025. 

212. The electrochemical equivalents of the most im- 
])ortant of the elements are given in the following table. 
They are calculated from the chemical equivalents in the 
l)receding column by simple proportion, taking as basis 
the above-named value for silver. Their reciprocals are 
the quantities of electricity required for depositing one 

* Rayleigh's detennination is 0111794; Kohlrausch's, '011183; 
Mascart's, -011156. See " PhU. Trans.," 1884, pp. 439, 458. 



180 



UNITS AND PHYSICAL CONSTANTS. [chap. 



gramme. The quantity of electricity required for deposit- 
ing the number of grammes stated in the column '' chemical 
equivalents" is the same for all the elements, namely, 
9634 C.G.S. units. 



Elemouts. 



Mectro-positive — 

Hydrogen, 

Potassium, 

Sodium, 

Gold, 

Silver, 

Copper (cupric), 

,, (cuprous), 

Mercury (mercuric),.. 
,, (mercurous), 

Tin (stannic), 

,, (stannous), 

Iron (ferric), 

„ (ferrous), 

Nickel, 

Zinc, 

Lea<l, 

Aluminium, 

Electro-negative — 

Oxygen, 

Chlorine, 

Iodine, , 

Bromine, , 

Nitrogen, 



Atomic 
Weight. 


• 

a 




I 


1 


1 


39-03 


1 


23 00 


1 


196-2 


3 


107-7 


1 


63-18 


2 


199-8 


1 
2 


It 
117-4 


1 
4 


55-88 


2 
3 


58-6 


2 
2 


64-88 


2 


206-4 


2 


27-04 


3 


15-96 


2 


35-37 


1 


126-54 


1 


79-76 


1 


14-01 


3 



Chemi- 
cal 
Equiva- 
lents. 



1 
39-03 
23-00 
65-4 

107-7 
31-59 
63-18 
99-9 

199-8 
29-35 
58-7 
18-63 
27-94 
29-3 
32-44 

103-2 
901 

7-98 

35-37 

126-54 

79-76 

4-67 



Electro- 
chemical 
equivalents 
orgrammes 
per unit of 
electricity. 



•0001038 

•004051 

•002387 

•006789. 

•01118 

•003279 

•006558 

•01037 

-02074 

-003046 

•006093 

-001934 

-002900 

-003042 

-003367 

•01071 

-000935 

-0008283 

-003671 

•013134 

•008279 

-0004847 



Recipro- 
cal or 
Electri- 
city per 
gramme. 



9634 
246-9 
418-9 
147-3 

89-45 
305-0 
152-5 

96-43 

48-22 
328-3 
164-1 
517-1 
344-8 
328-7 
297-0 

93-37 
1070 

1207 

272-4 
76-14 

120-8 
2063 



To find the equivalent of 1 coulomb, divide the above 
electrochemical equivalents by 10. 

To find the number of grammes deposited per hour by 
1 ampere, multiply the above electrochemical equivalents 
by 360. 



XI.] ELECTEICITY. 181 

213. Let the " chemical equivalents " in the above table 
be taken as so many grammes ; then, if we denote by H 
the amount of heat due to the whole chemical action 
which takes place in a battery cell during the consumption 
of one equivalent of zinc, the chemical energy which runs 
down, namely JH ergs, must be equal (if there is no 
wasteful local action) to the energy of the current pro- 
duced. But this is the product of the quantity of 
electricity 9634 by the electromotive force of the cell. 

TTT 

The electromotive force is therefore equal to -zrzr-* 

^ 9634 

In the tables of heats of combination which are in use 

among chemists, the equivalent of hydrogen is taken as 2 

grammes, and that of zinc as 64*88 or 65 grammes. The 

equivalent quantity of electricity will accordingly be 

9634 X 2, and the formula to be used for calculating the 

electromotive force of a cell will be w-q^^» 

In applying this calculation to Daniell's and Grove's 
cells, we shall employ the following heats of combination, 
which are given on page 614 of Watts' "Dictionary of 
Chemistry," vol. vii., and are based on Julius Thomson's 
observations : — 

Zn, 0, S03, Aq., 108,462 

Cu, O, S03, Aq., 54,225 

N202, 03, Aq., 72,940 

N202, 0, Aq., 36,340 

In Daniell's cell, zinc is dissolved and copper is set 
free, we have, accordingly, 

H = 108,462 - 54,225 = 54,237. 
In Grove's cell, zinc is dissolved and nitric acid is 



182 UNITS AND PHYSICAL CONSTANTS. [chap. 

changed into nitrous acid. The thermal value of this 
latter change can be computed from the third and fourth 
data in the above list, as follows : — 

72,940 is the thermal value of the action in which, by the 
oxidation of one equivalent of N^O^ and combination with 
water, two equivalents of NHO^ (nitric acid) are produced. 

36,340 is the thermal value of the action in which, by 
the oxidation of one equivalent of N^O^ and combination 
with water, two equivalents of NHO^ (nitrous acid) are 
produced. The difference 36,600 is accordingly the ther- 
mal value of the conversion of two equivalents of nitrous 
into nitric acid, and 18,300 is the value for the conversion 
of one equivalent In the present case the reverse changes 
take place. We have, therefore, 

H = 108,462 - 18,300 = 90,162. 

JH 

19268 
M82 X 108 for Danieirs cell. 

1-965x108 „ Grove's „ 

These are greater by from 2 to 8 per cent, than the direct 

determinations given in § 205. 

214. Examples in Electricity. 

1. Two conducting spheres, each of 1 centim. radius, are 
placed at a distance of r centims. from centre to centre, 
r being a large number; and each of them is charged 
with an electrostatic unit of positive electricity. With 
what force will they repel each other ? 

Since r is large, the charge may be assumed to be uni- 
formly distributed over their surfaces, and the force will 
be the same as if the charge of each were collected at its 

centre. The force will therefore be -, of a dyne. 



Taking J as 4*2 x lO'^, the value of ^^^^-tto ^^^ ^® 



XI.I ELECTRICITY. 183 

2. Two conducting spheres, eacli of 1 centim. radius, 
placed as in the preceding question, are connected one 
with each pole of a DanielFs battery (the middle of the 
battery being to earth) by means of two very fine wires 
whose capacity may be neglected, so that the capacity of 
each sphere when thus connected is sensibly equal to 
unity. Of how many cells must the battery consist that 

the spheres may attract each other with a force of - of a 

dyne, r being the distance between their centres in cen- 
tims. ? 

One sphere must be charged to potential 1 and the other 
to potential - 1. The number of cells required is 

? = 535. 

•00374 

3. How many DanielPs cells would be required to pro- 
duce a spark between two parallel conducting surfaces at 
a distance of '019 of a centim., and how many at a distance 
of -0086 of a centim. ? (See §§ 178, 184.) 

. 4-26 noQ 2-30 ^,^ 

•00374 ' -00374 

4. Compare the capacity denoted by 1 farad with the 
capacity of the earth. 

The capacity of the earth in static measure is equal to 
its radius, namely 6*37 x 10^. Dividing by v'^ to reduce 
to magnetic measure, we have '71 x 10"^^, which is 1 
farad multiplied by '71 x 10"', or is '00071 of a farad. 
A farad is therefore 1400 times the capacity of the earth. 

5. Calculate the resistance of a cell consisting of a 
plate of zinc, A square centim s. in area, and a plate of 
copper of the same dimensions, separated by an acid 



184 UNITS AND PHYSICAL CONSTANTS. [chap. 

solution of specific resistance 10^, the distance between 
the plates being 1 centim. 

Ana, — ) or — of an ohm. 

A A 

6. Find the heat developed in 10 minutes by the 
passage of a current from 10 Daniell's cells in series 
through a wire of resistance 10^^ (that is, 10 ohms), 
assuming the electromotive force of each cell to be 
I'l X 10®, and the resistance of each cell to be 10^. 
Here we have 

Total electromotive force = 1 '1 x P. 
Resistance in battery = 10^^. 



Resistance in wire = 10^^. 

M xlOQ 
2 X lO^o 



Current = l'^ "! LT = -55 x lO'i = -055. 



Heat developed in ) ^ (-0552) x IQio ^ 7.2004 

wire per second ) 4*2 x 10"^ ~ " * 

Hence the heat developed in 10 minutes is 43214 
gramme-degrees. 

7. Find the electromotive force between the wheels on 
opposite sides of a railway carriage travelling at the rate 
of 30 miles an hour on a line of the ordinary gauge 
[4 feet 8J inches] due to cutting the lines of force of 
terrestrial magnetism, the vertical intensity being '438. 

The electromotive force will be the product of the 
velocity of travelling, the distance between the rails, and 
the vertical intensity, that is, 

(44-7 X 30) (2-54 x 56-5) (-438) = 84,300 
electromagnetic units. 

This is about of a volt 

1 Ji\j\j 



XI.] ELECTRICITY. 185 

8. Find the electromotive force at the instant of passing 
tlie magnetic meridian, in a circular coil consisting of 300 
turns of wire, revolving at the rate of 10 revolutions per 
second about a vertical diameter ; the diameter of the 
coil being 30 centims., and the horizontal intensity of 
terrestrial magnetism being '1794, no other magnetic 
influence being supposed present. 

Self-induction can be left out of account, because the 
current is a maximum. 

The numerical value of the lines of force which go 
through the coil when inclined at an angle 6 to the 
meridian, is the horizontal intensity multiplied by the 
area of the coil and by sin 6 ; say nH-ira^ sin 6, where 
H = *1794, a =15, and 7i = 300. The electromotive 
force at any instant is the rate at which this quantity 
increases or diminishes ; that is, TiKira^ cos ^ . w, if w 
denote the angular velocity. At the instant of passing 
the meridian cos ^ is 1, and the electromotive force is 
nUTra^d), With 10 revolutions per second the value of cd 
is 27r X 10. 

Hence the electromotive force is 

•1794 X (3142)2 X 225 x 20 x 300 = 2-39 x 10^. 

This is about t^ of a volt. 

42 

190. To investigate the magnitudes of units of length, 
mass, and time which will fulfil the three following 
conditions : — 

1. The acceleration due to the attraction of unit mass 
at unit distance shall be unity. 

2. The electrostatic units shall be equal to the electro- 
magnetic units. 



186 UNITS AND PHYSICAL CONSTANTS. [chap. 

3. The density of water at 4" C. shall be unity. 

Let the 3 units required be equal respectively to L 
centims., M gi*ammes, and T seconds. 

We have in C.G.S. measure, for the acceleration due 
to attraction (§ 72), 

acceleration = ,— r^,, where C = 6*48 x 10"^ : 

(distance)^ 

and in the new system we are to have 

mass 



acceleration = 



(distance)^* 
Hence, by division, 

acceleration in C.G.S. units 
acceleration in new units 

_p mass in C.G.S. units (distance in new units)^ 
mass in new units (distance in C.G.S. units)^ ' 

, , . . L ^M 

that is, _ = (J . 
vxA^v x«, 1.2 

This equation expresses the first of the three conditions. 

The equation =^=v expresses the second, v denoting 

3 X 1010. 

The equation M = L^ expresses the third. 
Substituting L^ for M in the first equation, we find 

T= ^/^. Hence, from the second equation, 
and from the third, 

H'4)' 



XI.] ELECTRICITY. 187 

Introducing the actual values of C and v, we have 
approximately 

T = 3928, L= M78 x IQi*, M = 1-63 x 10*2 ; 

that is to say, 

The new unit of time will be about P 5 J™ ; 

The new unit of length will be about 118 thousand 

earth quadrants ; 
The new unit of mass will be about 2*66 x 10^* times 

the earth's mass. 

Electrodynamics, 

191. Ampere's formula for the repulsion between two 
elements of currents, when expressed in electromagnetic 
units, is 

cc' ds . ds' 



r 



2 



(2 sin a sin a cos - cos <x cos a ), 



where c, c' denote the strengths of the two currents ; 
dsy ds' the lengths of the two elements ; 

a, a' the angles which the elements make with 
the line joining them ; 
r the length of this joining line ; 
the angle between the plane of r, ds^ and 
the plane of r, ds'. 
For two parallel currents, one of which is of infinite 
length, and the other of length Z, the formula gives by 
integration an attraction or repulsion, 

21 , 

where D denotes the perpendicular distance between the 
currents. 



1 88 UNITS AND PHYSICAL CONSTANTS, [chap. xi. 

Eocample, 
Find the attraction between two parallel wires a metre 

long and a centim. apart when a current of - - is passing 

through each. 

Here the attraction will be sensibly the same as if 
one of the wires were indefinitely increased in length, 
and will be 

200/ 



!00/iy_2. 



that is, each wire will be attracted or repelled with a force 
of 2 dynes, according as the directions of the currents are 
the same or opposite. 



189 



OMISSION (to be added to § 63, p. 61). 

According to experiments by Quincke (Berlin Transac- 
tions, April 5, 1885) the following are the compressions 
due to the pressure of one atmosphere. They are ex- 
pressed in millionths of the original volume : — 

Compression 
in miilionths. 

^ '^ , 

at 0° C. at <• C. t. 

Glycerine, 25*24 25*10 19*00 

Rape oil (rttbol) 48*02 58*18 17*80 

AhnondoU 48*21 56*30 19*68 

OUveoil, 48*59 61*74 18*3 

Water, 50*30 45*63 22*93 

Bisulphide of carbon, 53 *92 63 *78 17 *00 

Oil of turpentine, 58*17 77*93 18*56 

Benzol from benzoic acitl, — 66*10 16*78 

Benzol, — 62*84 16*08 

Petroleum, 64*99 74*50 19*23 

Alcohol, 82*82 95*95 17*51 

Ether, 115*57 147*72 21*36 



CORRECTION (p. 84). 



Benoit's results on refraction of air will not appear in 
vol. v., but in a later volume. 



190 



SUGGESTION FOR WRITING DECIMAL 
MULTIPLES AND SUBMULTIPLES. 

Professor Newcomb has suggested, as a possible improve- 
ment in future editions of this work, the employment of 
powers of 1000 instead of powers of 10 as factors (a plan 
which corresponds with the usual division of digits into 
))eriods of 3 each)^ and the employment of the letter m in 
this connection to denote 1000. 

Thus, instead of 1*226 x 10^, we should write 122*6 m. 

1-006x107, „ 10 06 m2. 

•000 000 9, „ -9 7n-\ 

The plan appears to possess some advantages ; and if the 
symbol m for 1000 is not sufficiently self-explanatory, we 
might write 122*6 x lO^, 1006 x 10«, -9 x 10"^ We 
place the suggestion on record here that it may not be 
overlooked. 



191 



APPENDIX. 

^rst Report oftlte Committee for the Selection aiid Nomenclature 
of DynamicaX and ElectHcaZ UnitSy the Committee consisting of 
Sm W. Thomson, F.R.S., Professor G. C. Foster, F.R.S., 
Professor J. C. Maxwell, F.R.S., Mr. G. J. Stoney, 
F.R. S. ,* Professor Fleeming «Tenkin, F.R. S. , Dr. Siemens, 
F.R.S., Mr. F. J. Bramwell, F.R.S., and Professor 
Everett (Reporter). 

We consider that the most urgent portion of the task intrusted 
to us is that which concerns the selection and nomenclature of 
units of force and energy ; and under this head we are prepared 
to offer a definite recommendation. 

A more extensive and difficult part of our duty is the selection 
and nomenclature of electrical and magnetic units. Under this 
head we are prepared with a definite recommendation as regards 
iselection, but with only an interim recommendation as regards 
nomenclature. 

Up to the present time it has been necessary for every person 
who wishes to specify a magnitude in what is called ** absolute " 
measure, to mention the three fundamental units of mass, length, 
and time which he has chosen as the basis of his system. This 
necessity will be obviated if one definite selection of three funda- 
mental units be made once for all, and accepted by the general 
-consent of scientific men. We are strongly of opinion that such 
a selection ought at once to be made, and to be so made that 
there will be no subsequent necessity for amending it. 

We think that, in the selection of each kind of derived unit, all 
arbitrary multiplications and divisions by powers of ten, or other 
factors, must be rigorously avoided, and the whole system of 

* Mr. Stoney objected to the selection of the centimetre as the unit of length. 



192 APPENDIX. 

fundamental units of force, work, electrostatic, and electromag- 
netic elements must be fixed at one common level — that level, 
namely, which is determined by direct derivation from the three 
fundamental units once for all selected. 

The carrying out of this resolution involves the adoption of 
some units which are excessively large or excessively small in 
comparison with the magnitudes which occur in practice ; but a 
remedy for this inconvenience is provided by a method of denoting 
decimal multiples and sub-multiples, which has already been 
extensively adopted, and which we desire to recommend for. 
general use. 

On the initial question of the particular units of mass, length, 
and time to be recommended as the basis of the whole system, a 
protracted discussion has been carried on, the principal point 
discussed being the claims of the gramme, the metre^ and the 
second, as against the gramme, the centimetre^ and the second, — 
the former combination having an advantage as regards the 
simplicity of the name mttrey while the latter combination has 
the advantage of making the unit of mass practically identical 
with the mass of unit- volume of water — in other words, of making 
the value of the density of water practically equal to unity. We 
are now all but unanimous in regarding this latter element of 
simplicity as the more important of the two ; and in support of 
this view we desire to quote the authority of Sir W. Thomson, 
who has for a long time insisted very strongly upon the necessity 
of employing units which conform to this condition. 

We accordingly recommend the general adoption of the Centi- 
metre, the Gramme, and the Second as the three fundamental 
units ; and until such time as special names shall be appropriated 
to the units of electrical and magnetic magnitude hence derived, 
we recommend that they be distinguished from ** absolute" units 
otherwise derived, by the letters "C.G.S." prefixed, these being 
the initial letters of the names of the three fundamental units. 

Special names, if short and suitable, would, in the opinion of a 
majority of us, be better than the provisional designations ** C.G.S. 
unit of . . . ." Several lists of names have already been 
suggested ; and attentive consideration will be given to any further 



APPENDIX. 193 

suggestions which we may receive from persons interested in 
electrical nomenclature. 

The "ohm," as represented by the original standard coil, is 
approximately 10» C.G.S. units of resistance; the "volt" is 
approximately lO^ C.G.S. units of electromotive force ; and the 

" farad " is approximately -^ of the C.G.S. unit of capacity. 

For the expression of high decimal multiples and sub-multiples, 
we recommend the system introduced by Mr. Stoney, a system 
which has already been extensively employed for electrical pur- 
poses. It consists in denoting the exponent of the power of 
10, which serves as multiplier, by an appended cardinal num- 
ber, if the exponent be positive, and by a prefixed ordinal number 
if the exponent be negative. 

Thus 10^ grammes constitute a gramme-nine; — ^ of a gramme 

10^ 

constitutes a ninth-gramme; the approximate length of a quadrant 
of one of the earth's meridians is a metre-seven, or a centimetre- 
nine. 

For multiplication or division by a million, the prefixes mega* 
and micro may conveniently be employed, according to the present 
custom of electricians. Thus the megohm is a million ohms, and 
the microfarad is the millionth part of a farad. The prefix mega 
is equivalent to the affix six. The prefix micro is equivalent to 
the prefix sixth. 

The prefixes ktlo, hecto, deca, deci, centi, milli can also be em- 
ployed in their usual senses before all new names of units. 

As regards the name to be given to the C.G.S. unit of force, we 
recommend that it be a derivative of the Greek diJva/us. The form 
dyrumny appears to be the most satisfactory to etymologists. 
Dynam is equally intelligible, but awkward in sound to English 
ears. The shorter form, dyne, though not fashioned according to 
strict rules of etymology, will probably be generally preferred in 
this country. Bearing in mind that it is desirable to construct a 
system with a view to its becoming international, we think that 

* Before a vowel, either tmq or megcUt as euphony may suggest, may be 
employed instead of mega. 

N 



194 APPENDIX. 

the termmation of the word should for the present be left an open 
question. But we would earnestly request that, whichever form of 
the word be employed, its meaning be strictly limited to the unit of 
force of the C.G.S. system — that is to say, the force which, acting 
upon a gramme of matter for a second, generates a velocity of a 
centimetre per second. 

The C.G.S. unit of work is the work done by this force working 
through a centimetre; and we propose to denote it by some deriva- 
tive of the Greek fpyov. The forms ergon, ergal, and erg have 
been suggested; but the second of these has been used in a 
different sense by Clausius. In this case also we propose, for 
the present, to leave the termination unsettled; and we request 
that the word ergon, or erg, be strictly limited to the C.G.S. unit 
of work, or what is, for purposes of measurement, equivalent to 
this, the C.G.S. unit of energy, energy being measured by the 
amount of work which it represents. 

The C.G.S. unit of power is the power of doing work at the rate 
of one erg per second ; and the power of an engine, under given 
conditions of working, can be specified in ergs per second. 

For rough comparison with the vulgar (and variable) units 
based on terrestrial gravitation, the following statement will be 
useful : — 

The weight of a gramme, at any part of the earth's surface, is 
about 980 dynes, or rather less than a kilodyne. 

The weight of a kilogramme is rather less than a megadyne, being 
about 980,000 dynes. 

Conversely, the dyne is about 1 *02 times the weight of a milli- 
gramme at any part of the earth's surface ; and the megadyne is 
about 1 *02 times the weight of a kilogramme. 

The kilogrammetre is rather less than the ergon-eight, being 
about 98 million ergs. 

The gramme-centimetre is rather less than the kilerg, being 
about 980 ergs. 

For exact comparison, the value of g (the acceleration of a body 
falling in vacuo) at the station considered must of course be 
known. In the above comparison it is taken as 980 C.G.S. units 
of acceleration. 



APPENDIX. 195 

One horse-potoer is about three quarters of an erg-ten per second. 
More nearly, it is 7 '46 erg-nines per second, and one force-de-cheval 
is 7 '36 erg-nines per second. 

The mechanical equivalent of one gramme-degree (Centigrade) 
of heat is 41*6 megalergs, or 41,600,000 ergs. 



Second Report oftlie Committee for the Selection and Nomeiidature 
of Dynamical and Electrical Units, the Committee consisting of 
Professor Sir W. Thomson, F.R.S., Professor G. C. 
Foster, F.R.S., Professor J. Clerk Maxwell, F.R.S., 
G. J. Stone Y, F.R.S., Professor Fleeming Jenkin, F.R.S., 
Dr. C. W. Siemens, F.R.S., F. J. Bramwell, F.R.S., 
Professor W. G. Adams, F.R.S., Professor Balfour 
Stewart, F.R.S., and Professor Everett (Secretary). 

The Committee on the Nomenclature of Dynamical and Electrical 
Units have circulated numerous copies of their last year's Report 
among scientific men both at home and abroad. 

They believe, however, that, in order to render their recom- 
mendations fully available for science teaching and scientific 
work, a full and popular exposition of the whole subject of 
physical units is necessary, together with a collection of examples 
(tabular and otherwise) illustrating the application of systematic 
units to a variety of physical measurements. Students usually 
find peculiar difficulty in questions relating to units; and even the 
experienced scientific calculator is glad to have before him con- 
crete examples with which to compare his own results, as a 
security against misapprehension or mistake. 

Some members of the Committee have been preparing a small 
volume of illustrations of the C.G.S. system [Centimetre-Gramme- 
Second system] intended to meet this want. 

[The first edition of the present work is the volume of illustra- 
tions here referred to.] 



196 



INDEX. 



The numbers refer to the pages. 



Acceleration, 25. 
Acoustics, 70-74. 
Adiabatic compressioD, 125. 
Air, collected aata for, 129. 

, density of, 43. 

, expansion of, 99. 

, specific heat of, 94, 123. 

, tnermal conductivity of, 

108. 
Ampere as unit, 151-153. 
Ampere's formula, 187. 
Aqueous vapour, pressure of, 

100-102. 

, density of, 102. 

Astronomy, 65-69. 
Atmosphere, standard, 42, 43. 

, its density upwards, 47. 

Atomic weights, 180. 
Attraction, constant of, 67. 

at a point, 17. 

Angle, 16. 
, solid, 17. 

Barometer, correction for capil- 
larity, 51. 

Barometric measurements of 
heights, 47. 

pressure, 42. 

Batteries, 166-168, 172, 181. 

Boiling points, 98. 

of water, 100-102. 



Boyle's law, departures from, 99. 
Bullet, melted Dy impact, 31. 

Candle, standard, 86. 

Capacity, electrical, 141-143. 

, specific inductive, 147-150. 

, thermal, 87-95. 

Capillarity, 49-51. 

Carcel, 86. 

Cells, 166-168, 172, 181. 

Centimetre, reason for selecting, 
23, 192. 

Centre of attraction, strength 
of, 17. 

Centrifugal force, 32. 

at equator, 34. 

C.G.S. system, 23, 192. 

Change of volume in evapora- 
tion, 97. 

in melting, 96, 97. 

Change-ratio, 9. 

Chemical action, heat of, 122. 

equivalents, 180. 

Clark's standard cell, 168. 

Cobalt, magnetization of, 136. 

Coil, revolving, 185. 

Combination, heat of, 122, 181. 
Combustion, heat of, 122. 
Common scale needed, 22. 
Comparison of standards (IlVench 
and English), 1, 2. 



INDEX. 



197 



Compressibility of liquids, 60, 
61, 189. 

of solids, 61-63. 

Compression, adiabatic, 125. 
Conductivity (thermal) defined, 

103. 
, thermometric and calori- 

metric, 105. 

of air, 108. 

of liquids, 116, 117. 

of various solids, 109-116. 

Congress of electricians, 153. 
Contact electricity, 168-172. 
Cooling, 117-120. 
Current, beat generated by, 

143, 166. 

, unit of, 141, 142, 151, 153. 

Curvature, dimensions of, 17, 18. 

DanieU's cell, 166, 167, 173, 181. 

Day, sidereal, 66. 

Decimal multiples, 24, 190, 193. 

Declination, magnetic, at Green- 
wich, 138. 

Densities, table of, 40. 

of gases, 44. 

of water, 38-39. 

Density as a fundamental unit, 
146. 

Derived units, 5, 6. 

Dew-point from wet and dry 
bulb, 102. 

Diamagnetic substances, 133. 

Diamond, specific heat of, 90. 

Diffusion, coefficient of, 105-108. 

Diffusivity (thermal), 105. 

Dimensionid equations, 9, 34-37. 

Dimensions, 7-9, 34-37. 

Dip at Greenwich, 138. 

Dispersive powers of gases, 83- 
85. 

of solids and liquids, 

77-82. 

Diversity of scales, 22. 

"Division," extended sense of, 
10. 

Doable refraction, 81. 



Dynamics, 15-17. 
Dyne, 27, 193. 

Earth as a magnet, 136. 

, size, figure, and mass of, 

65. 

Elasticity, 52-64. 

, effected by heat of com- 
pression, 127. 

Electric units, tables of their 
dimensions, 143, 146. 

Electricity, 140-148. 

Electrochemical equivalents, 
180. 

Electrodynamics, 187. 

Electromagnetic units, 142. 

Electromotive force, 166-172, 
180182. 

Electrostatic units, 140. 

Emission of heat, 117-120. 

Energy, 29. 

, dimensions of, 16. 

Equations, dimensional, 9, 34-37. 

, physical, 12. 

Equivalent, mechanical, of heat, 
120. 

Equivalents, electrochemical, 
180. 

Erg, 29, 194. 

Evaporation, change of volume 
in, 97. 

Examples in electricity, 1 82-1 86. 

in theory of units, 12-15, 

34-37. 

in magnetism, 134-137. 

Expansion of gases, 99. 

of mercury, 128, 129. 

of various substances, 128, 

129. 

Extended sense of ''multiplica- 
tion " and "division, 10. 

Farad, 151-153. 

compared with earth, 183. 

Field, intensit^r of, 131. 
Films, tension in, 49, 50. 
, thickness of, 50, 51. 



198 



INDEX. 



Foot-pound and foot-poundal, 
30. 

Force, 27. 

, dimensions of, 15. 

at a point, 17. 

, various units of, 4. 

Freezing-point, change with 
pressure, 124. 

Frequencies of luminous vibra- 
tions, 77. 

Fundamental units, 6. 

, choice of, 19. 

reduced to two, 68. 

Gases, densities of, 44. 

, expansion of, 99. 

, indices of refraction of, 82. 

, inductive capacities of, 

151. 

, two specific heats of, 123. 

Gauss's expression for magnetic 

potential, 138. 

pound-magnet, 134. 

units of intensity, 137. 

Geometrical quantities, dimen- 
sions of, 15-18. 
Gottingen, total intensity at, 

137. 
Gramme-degree (unit of heat), 

88. 
Gravitation in astronomy, 67. 
Gravitation measure of force 

and work, 28, 30. 
Gravity, terrestrial, 25-27. 
Greenwich, magnetic elements 

at, 138. 
Grove's ceU, 167, 172, 181. 

Heat, 87-130. 

generated by current, 143, 

166. 
, mechanical equivalent of, 

120. 

* of combination, 122, 181. 

of compression, 125. 

., unit of, 87, 88. 



— , various units of, 3. 



Height, measured by barometer, 

47. 
Homogeneous atmosphere,45-47. 
Horse-power, 30. 
Hydrostatics, 38-51. 
Hypsometric table of boiling 

points, 100. 

Ice, specific gravity of, 96, 125. 

, specific heat of, 92. 

, electrical resistance of, 

163. 

Indices of refraction, 77-85. 

related to induc- 
tive capacities, 147, 148. 

Inductive capacity, 147-150. 

Induction, magnetic, coefficient 
of, 133. 

Insulators, resistance of, 164, 
165. 

Interdiffusion, 106-108. 

Joule's equivalent, 120. 

Kilogramme and pound, 2. 
Kinetic energy, 29. 
KupfFer's determination of den- 
sity of water, 38. 

Large numbers, mode of expres- 
sing, 24, 190, 193. 
Latent heats, 95-98. 
Latimer Clark's cell, 168. 
Light, 75-86. 

, velocity of, 75, 76. 

, wave-lengths of, 76. 

Magnetic elements at Green- 
wich, 138. 

susceptibility, 133. 

units, 131, 132. 

Magnetism, 131-139. 

, terrestrial, 136-138. 

Magneto-optic rotation, 139. 

Magnetization, intensity of, 
132, 133. 

Mass, standards of, 20. 



INDEX. 



199 



Mechanical equivalent of heat, 

120. 
quantities, dimensions of, 

15. 

units, 27. 

Mega, as prefix, 42, 193. 
Melting points, 95-97. 
Metre and yard, 1. 
Micro as prefix, 193. 
Microfarad, 151. 
Moment of couple, 16. 

of inertia, 16. 

of magnet, 132. 

of momentum, 16. 

Momentum, 15. 

Moon, 66. 

* 'Multiplication, "extended sense 

of, 10. 

Neutral point (thermoelectric), 

175. 
Newcomb on decimal multiples, 

190. 
Nickel, magnetization of, 136. 
Numerical value, 5. 

Ohm as unit, 151-154. 

earth quadrant per second, 

155. 

-, ** legal," 153. 



Optics, 75-86. 

Paramagnetic substances, 133. 

Pendulum, seconds', 25, 26. 

** Per," meaning of, 10. 

Physical deductions from di- 
mensions, 34-37. 

Platinoid, 160. 

Platinum, specific heat of, 90. 

Poisson's ratio, 62. 

Potential, electric, 140. 

, magnetic, 131. 

Poundal, 28. 

Powers of ten as factors, 24, 
190, 193. 

Pressure, dimensions of, 17. 

of liquid columns, 42. 



Pressure, various units of, 3, 4. 
Pressures of vapours, 101. 
Pressure-height, 46. 

Quantity of electricity, 140, 142. 

Radian, 16. 

Radiation, 117-120. 

Ratios of two sets of electric 
units, 143. 

Refraction, indices of, 77-85. 

Reports of Units Committee, 
191-195. 

Resilience, 54. 

affected by heat of com- 
pression, 126. 

Resistance, electrical, 158-166. 

of a cell, 161, 183. 

of wires, 165, 166. 

Rigidity, simple, 55. 

Rotating coil, 185. 

Saturation, magnetic, 133-136. 
Shear, 55-58. 
Shearing stress, 58-60. 
Siemens^ unit, 152-154. 
Soap films, 50. 
Sound, faintest, 74. 

, velocitv of, 70-73. 

Spark, length of, 155-157. 
Specific gravities, 40. 

, heat, 88-95. 

, two, of gases, 123. 

, inductive capacity, 147- 

150. 
Spring balance, 31. 
Standards, French and English, 

1,2. 

of length, 21. 

of mass, 20. 

of time, 21. 

Steam, pressure and density of, 

100-102. 
, total and latent heat of, 

98. 
Stoney's nomenclature for 

multiples, 193. 



200 



INDEX. 



Strain, 52, 53, 55-58. 

, dimensions of, 53. 

Stress, 52-54, 58-60. 

, dimensions of, 54. 

Strings, musical, 73. 
Sun's distance and parallax, 66. 
Supplemental section on dimen- 
sions, 34-37. 
Surface-conduction, 1 17-120. 
Surface-tension, 49, 50. 

Telegraphic cables, resistance 
of, 160. 

Tenacities, table of, 64. 

Tensions of liquid surfaces, 49, 
50. 

Thermodynamics, 120-128. 

Thermoelectricity, 172-178. 

Time, standard of, 21. 

Tortuosity, 17. 

Two fundamental units suffici- 
ent, 68. 

Unit, 5. 

Units, derived, 5, 6. 

, dimensions of, 7-9. 

, special problems on, 69, 

185. 

Vapours, pressure of, 101. 



Velocity, 6, 9. 

of Ught, 75, 76. 

of sound, 70-73. 

, various units of, 2. 

Vibrations per second of light, 

77. 
Volt, 151-153. 
Volume, by weighing in water, 

40. 

of a gramme of gas, 44. 

, unit of, 5. 

Volume resilience, 55, 60-63, 189. 

Water, compressibility of, 60, 
61, 189. 

, densitjr of, 38, 39. 

, expansion of, 39. 

, specific heat of, 87, 88. 

, weighing in, 40. ii 

Watt (rate of working), 4, 30. 
Weight, force, and mass, 27, 28. 

, standards of, 20-2. 

Wires (Imperial gauge), 165,166. 
Work, 29, 30, 3, 4. 

, dimensions of, 16. 

done by current, 143. 

Working, rate of, 30, 3, 4. 

Year, sidereal and tropical, 66. 
Young's modulus, 55. 



PBINTED BT ROBERT MACLEHOSE, UNIVERSITY PRESS, QLASaOW. 



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Aldis. — THE GREAT GIANT ARITHMOS. A most Elementa 
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Brook-Smith (J.).— ARITHMETIC IN THEORY AN 
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alton. — RULES AND EXAMPLES IN ARITHMETIC. By 
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New Edition. iSmo. zr. 6d, 

[Answers to the ExampJesare appended. 

OCk. — ^ARITHMETIC FOR SCHOOLS. By Rev. J. B. I OCK, 
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Smith. — Works by the Rev. Barnard Smith, M. A. {i 
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EASY LESSONS IN ARITHMETIC, combining Exercises 
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ALGEBRA. 

Dalton. — RULES AND EXAMPLES IN ALGEBRA. Byrt 
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\* A Key to Part I. is now in the press, 

Jones and Cheyne. — algebraical exercises. Pro-| 

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H. CHEYNE, M.A., F.K.A.S., Mathematical Masters of West- 
minster School. New lidition. i8mo. 2j. td, \ 

i 

Hall and Knight.— ELEMENTARY ALGEBRA for; 

schools. By II. S. Hall, M.A., formerly Scholar of Christ's 
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ALGEBRAICAL EXERCISES AND EXAMINATION PAPERS. 
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HIGHER ALGEBRA FOR SCHOOLS. By the same Authors 
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land, and Fellow and Senior Bursar of St. Peter's College, Cam- 
bridge. New Edition, carefully Revised. Crown 8vo. loj. 6<^. 
nith (Charles). — Works by Charles Smith, M.A., Fellow 

and Tutor of Sidney Sussex College, Cambridge. 
ELEMENTARY ALGEBRA. Globe 8vo. fr. 6^. 
n this work the author has endeavoured to explain the principles of Algebra in as 
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; explanations and proofs of the fundamental f perations and rules, 
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Odhunter. — Works by L Todhunter, M.A., F.R.S., D.Sc, 

late of St. John's College, Cambridge. 
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EUCLID & ELEMENTARY GEOMETRY. 

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athbertson. — EUCLIDIAN geometry. By Francis 
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agles.— CONSTRUCTIVE GEOMETRY OF PLANE 
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ELEMENTS FOR THE USE OF SCHOOLS. By H. S 
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Halsted.— THE ELEMENTS OF GEOMKTRY. By GeoM 
BRUCE HALSTED, Pr fessor of Pure and Applied MatU 
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Kitchener. — a geometrical NOTE-BOOK, contalniB! 
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shire. New Edition. 4to. zr. 

Mault.— NATURAL GEOMETRY: an Introduction to tb 
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Syllahus of Plane Geometry (corresponding to Eodii 

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Todhunter. — the elements of EUCLID. For the Us 
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of St. John's College, Cambridge. New Edition. iSmo. y. & 
KEY TO EXERCISES IN EUCLID. Crown 8vo. 6j. W. 

Wilson (J. M.). — ELEMENTARY GEOMETRY. BOOK: 
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ON THE ALGEBRAICAL AND NUMERICAL THEORY 

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Icxander (T.).— elementary applied mechanics. 

Being the simpler and more practical Cases of Stres*^ and Strain 
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[pole. — THE CALCULUS OF FINITE DIFFERENCES. 

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unbridge Senate-House Problems and Riders, 
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iristie.— A collection of elementary test- 

QUESTIONS IN PURE AND MIXED MATHEMATICS; 
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ClausiuS. — MECHANICAL THEORY OF HEAT. By RJ 

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Day (R. E.) -electric light arithmetic. By R. E. 

Day, M.A., Evening Lecturer in Experimental Physics at King's 
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Drew.— GEOMETRICAL TREATISE ON CONIC SECTIONS. 
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Dyer. — EXERCISES IN ANALYTICAL GEOMETRY. Com- 
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Eagles.— CONSTRUCTIVE GEOMETRY OF PLANE 
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Edgar (J. H.) and Pritchard (G. S.). — NOTE-BOOK ON 

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[In thepresi. 

Ferrers. — Works by the Rev. N. M. Ferrers, M.A., Master of 
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AN ELEMENTARY TREATISE ON TRILINEAR CO- 
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AN ELEMENTARY TREATISE ON SPHERICAL HAR 
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AN ELEMENTARY TREATISE ON CURVE TRACING. By 

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SOLID GEOMETRY. A New Edition, revised and enlarged, of 

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Greenhill. — differential and integral cal- 

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Xialsted. — the elements of geometry. By George 
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Kemming. — an elementary treatise on the 

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Ibbetson.— THE mathematical theory of per- 
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Merriman. — a TEXT BOOK OF THE METHOD OF LEAST 
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dillar. — ^ELEMENTS OF DESCRIPTIVE GEOMETRY. By 
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tf ilne. — WEEKLY PROBLEM PAPERS. With Notes intended 
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?irie. — LESSONS ON RIGID DYNAMICS. By the Rev. G. 
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12 MACMILLAN'S EDUCATIONAL CATALOGUE. 






Robinson, — ^treatise on marine surveying. Pre- 

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CoMTBNTS. — Symbols u.sed in Charts and Surveying— The Construction and Ua 
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enlarged Edition. Two Vols. 8vo. VoL I. — Elementary Parts. 

145. Vol. II. — The Advanced Parts. 14s. . 

STABILITY OF A GIVEN STATE OF MOTION, PAR- I 

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CONIC SECTIONS. Second Edition. Crown 8vo. 7s. 6d, 
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Todhunter. — Works by I. Todhunter, M.A., F.R.S., D.Sc, 

late of St. John's College, Cambridge. 

" Mr. Todhunter is chiefly known to students of Mathentiatics as the author of » 
series of admirable mathematical text-books, which possess the rare Qualities of bemg 
clear in style and absolutely free from mistakes, typographical and other."— 
Saturday Review. 

TRIGONOMETRY FOR BEGINNERS. With numerous 

Example ^\ New Edition. i8mo. 2s. 6J, 



] 






,' < 



MATHEMATICS. 13 



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SCIENCE. 15 



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SCIENCE, 19 



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V 2 



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22 MACMILLAN'S EDUCATIONAL CATALOGUE. 

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SCIENCE. 23 



Poster and Langley.— a COURSE OF ELEMENTARY 

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